First-Passage Statistics of Extreme Values

Transcription

1 First-Passage Statistics of Extreme Values Eli Ben-Naim Los Alamos National Laboratory with: Paul Krapivsky (Boston University) Nathan Lemons (Los Alamos) Pearson Miller (Yale, MIT) Talk, publications Random Graph Processes, Austin TX, May 10, 2016

2 Plan I. Motivation: records & their first-passage statistics as a data analysis tool II. Incremental records: uncorrelated random variables III.Ordered records: uncorrelated random variables IV. Ordered records: correlated random variables

3 I. Motivation: records & their first-passage statistics as a data analysis tool

4 Record & running record Record = largest variable in a series X N = max(x 1,x 2,...,x N ) Running record = largest variable to date X 1 apple X 2 apple applex N Independent and identically distributed variables Z 1 0 dx (x) =1 Statistics of extreme values Feller 68 Gumble 04 Ellis 05

5 Average number of running records Probability that Nth variable sets a record P N = 1 N Average number of records = harmonic number M N = N Grows logarithmically with number of variables M N ' ln N + = Behavior is independent of distribution function Number of records is quite small

6 Time series of massive earthquakes M= year M= year M= year 1770 M>7 events Magnitude Annual # / ~13, ~130, ~1,300,000 Are massive earthquakes correlated?

7 Records in inter-event time statistics M N t 1 t 2 t 3 t 4 Count number of running records in N consecutive events 9 8 M>5 (37,000 events) 7 M>7 (1,770 events) 6 1+1/2+1/ /N records indicate inter-event times uncorrelated N attribute deviation to aftershocks massive earthquakes are random EB & Krapivsky PRE 13

8 First-passage processes Process by which a fluctuating quantity reaches a threshold for the first time First-passage probability: for the random variable to reach the threshold as a function of time. Total probability: that threshold is ever reached. May or may not equal 1 First-passage time: the mean duration of the first-passage process. Can be finite or infinite S. Redner, A Guide to First-Passage Processes, 2001

9 II. Incremental records: uncorrelated random variables

10 Marathon world record Year Athlete Country Record Improvement 2002 Khalid Khannuchi USA 2:05: Paul Tergat Kenya 2:04:55 0: Haile Gebrsellasie Ethiopia 2:04:26 0: Haile Gebrsellasie Ethiopia 2:03:59 0: Patrick Mackau Kenya 2:03:38 0: Wilson Kipsang Kenya 2:03:23 0:15 Incremental sequence of records every record improves upon previous record by yet smaller amount Are incremental sequences of records common? source: wikipedia

11 Incremental records y 4 3 y 2 y 1 Incremental sequence of records every record improves upon previous record by yet smaller amount random variable = {0.4, 0.4, 0.6, 0.7, 0.5, 0.1} latest record = {0.4, 0.4, 0.6, 0.7, 0.7, 0.7} latest increment = {0.4, 0.4, 0.2, 0.1, 0.1, 0.1} What is the probability all records are incremental?

12 Probability all records are incremental N -0.3 Earthquake data 10-1 simulation N S S N N S N N = Power law decay with nontrivial exponent Problem is parameter-free Miller & EB JSTAT 13

13 Uniform distribution 1/(N + 1) / 1/N The variable x is randomly distributed in [0:1] (x) = 1 for 0 apple x apple 1 Probability record is smaller than x R N (x) =x N Average record A N = N N +1 =) 1 A N ' N 1 Distribution of records is purely exponential

14 Distribution of records Probability a sequence is incremental and record < x One variable G N (x) =) S N = G N (1) G 1 (x) =x =) S 1 =1 x 2 = x 1 x 2 =2x 1 x/2 Two variables x 2 x 1 >x 1 G 2 (x) = 3 4 x2 =) S 2 = 3 4 In general, conditions are scale invariant x x ax x Distribution of records for incremental sequences G N (x) =S N x N S 1 =1 S 2 =3/4 S 3 = 47/72 Distribution of records for all sequences equals x N Statistics of records are standard Fisher-Tippett 28 Gumbel 35

15 Scaling behavior G N (x)/g N (1) N=1 N=2 N=3 N=4 x x 2 x 3 x x Distribution of records for incremental sequences G N (x)/s N = x N =[1 (1 x)] N e s Scaling variable s =(1 x)n Exponential scaling function

16 Distribution of increment+records Probability density SN(x,y)dxdy that: 1. Sequence is incremental 2. Current record is in range (x,x+dx) 3. Latest increment is in range (y,y+dy) with 0<y<x Gives the probability a sequence is incremental Recursion equation incorporates memory S N+1 (x, y) =xs N (x, y)+ old record holds Z x a new record is set Evolution equation includes integral, has memory S N (x, y) N S N = Z 1 0 dx Z x = (1 x)s N (x, y)+ 0 dy S N (x, y) y y Z x y dy 0 S N (x y, y 0 ) y dy 0 S N (x y, y 0 )

17 Scaling transformation Assume record and increment scale similarly y 1 x N 1 Introduce a scaling variable for the increment s =(1 x)n and z = yn Seek a scaling solution S N (x, y) =N 2 S N (s, z) Eliminate time out of the master equation 2 + s + s s + z z (s, z) = Z 1 z dz 0 (s + z,z 0 ) Reduce problem from three variables to two

18 Factorizing solution Assume record and increment decouple Substitute into equation for similarity solution 2 + s + s s + z z (s, z) =e s f(z) (s, z) = Z 1 z dz 0 (s + z,z 0 ) First order integro-differential equation zf 0 (z)+(2 )f(z) =e z Z 1 Cumulative distribution of scaled increment z f(z 0 )dz 0 g(z) = Z 1 z f(z 0 )dz 0 Convert into a second order differential equation zg 00 (z)+(2 )g 0 (z)+e z g(z) =0 g(0) = 1 g 0 (0) = 1/(2 ) Reduce problem from two variable to one

19 Distribution of increment Assume record and increment decouple zg 00 (z)+(2 )g 0 (z)+e z g(z) =0 Two independent solutions g(0) = 1 g 0 (0) = 1/(2 ) g(z) =z 1 and g(z) = const. as z 1 The exponent is determined by the tail behavior = The distribution of increment has a broad tail P N (y) N 1 y Increments can be relatively large problem reduced to second order ODE 2

20 Numerical confirmation Monte Carlo simulation versus integration of ODE g g(0) = 1 g 0 (0) = 1/(2 ) theory simulation z hszi hsihzi 1 Increment and record become uncorrelated

21 Recap II Probability a sequence of records is incremental Linear evolution equations (but with memory) Dynamic formulation: treat sequence length as time Similarity solutions for distribution of records Probability a sequence of records is incremental decays as power-law with sequence length Power-law exponent is nontrivial, obtained analytically Distribution of record increments is broad First-passage properties of extreme values are interesting

22 III. Ordered records: uncorrelated random variables

23 x n y n Ordered records Motivation: temperature records: Record high increasing each year X n Y n Two sequences of random variables n {X 1,X 2,...,X N } and {Y 1,Y 2,...,Y N } Independent and identically distributed variables Two corresponding sequences of records x n = max{x 1,X 2,...,X n } and y n = max{y 1,Y 2,...,Y n } Probability SN records maintain perfect order x 1 >y 1 and x 2 >y 2 and x N >y N

24 Two sequences Survival probability obeys closed recursion equation Solution is immediate S N = S N 1 1 S N = 2N Large-N: Power-law decay with rational exponent N 1 2N 2 2N S N ' 1/2 N 1/2 Universal behavior: independent of parent distribution identical to Sparre Andersen 53

25 Ordered random variables Probability PN variables are always ordered X 1 >Y 1 and X 2 >Y 2 and X N >Y N Exponential decay P N =2 N Ordered records far more likely than ordered variables Variables are uncorrelated Records are strongly correlated: each record remembers entire preceding sequence Ordered records better suited for data analysis

26 Three sequences x n y n z n X n Y n Z n n Third sequence of random variables S N simulation theory σ N N x n >y n >z n n =1, 2,...,N Probability SN records maintain perfect order Power-law decay with nontrivial exponent? S N N with =1.3029

27 leader Rank of median record median laggard {z } j Closed equations for survival probability not feasible Focus on rank of the median record Rank of the trailing record irrelevant Joint probability PN,j that (i) records are ordered and (ii) rank of the median record equals j Joint probability gives the survival probability S N = NX j=1 P N,j

28 Closed recursion equations {z } j Closed recursion equations for joint probability feasible P N+1,j = N +2 j 3N +3 3N +2 j 3N +3 3N +1 3N +2 3N +1 3N +2 3N +2 j (3N + 3)(3N + 2)(3N + 1) 3N +2 j (3N + 3)(3N + 2)(3N + 1) j 3N j 3N +1 P N,j j j 3N +1 P N,j 1 X (3N k)p N,k N+1 The survival probability for small N k=j N+1 X k=j kp N,k 1 O(N 0 ) O(N 1 ) O(N 1 ) O(N 2 ) no new records new leading records new median records two new records N S N (3N) S N

29 Key observation P j,n S N N=10 2 N=10 3 N= j Rank of median record j and N become uncorrelated

30 Asymptotic analysis Rank of median record j and N become uncorrelated P N,j ' S N p j as N 1 Assume power law decay for the survival probability S N N The asymptotic rank distribution is normalized 1X j=1 Rank distribution obeys a much simpler recursion p j =(j + 1) p j Scaling exponent p j =1 j 3 p 1 j 1 3 1X k=j p k is an eigenvalue

31 The rank distribution EB & Krapivsky PRE 15 First-order differential equation for generating function (3 z) dp (z) dz + P (z) 1 1 z 3 z = z 1 z P (z) = X j 1 p j z j+1 Solution P (z) = r 1 z z 3 z 3 z Z z 0 1/2 du (3 u) (1 u) 3/2 u 1 Behavior near z=3 gives tail of the distribution p j j 1/2 3 j Behavior near z=1 gives the scaling exponent 2F 1 1 2, 1 2 ; 3 2 ; 1 2 =0 =) = Three sequences: scaling exponent is nontrivial

32 Multiple sequences Probability SN that m records maintain perfect order Expect power-law decay with m-dependent exponent S N N Lower and upper bounds 0 apple m m apple m m σ m Exponent grows linearly with number of sequences (up to a possible logarithmic correction) m ' m In general, scaling exponent is nontrivial

33 Family of ordering exponents One sequence always in the lead: 1>rest Two sequences always in the lead: 1>2>rest A N N m m =1 B N N m 2F 1 1 m 1 m 1, m 2 m 1 ; 2m 3 m 1 ; 1 m 1 =0 Three sequences always in the lead: 1>2>3>rest δ γ β α σ C N N m m /2 1/2 3 2/ / / / m

34 Recap III Probability multiple sequences of records are ordered Uncorrelated random variables Survival probability independent of parent distribution Power-law decay with nontrivial exponent Exact solution for three sequences Scaling exponent grows linearly with number of sequences Key to solution: statistics of median record becomes independent of the sequence length (large N limit) Scaling methods allow us to tackle combinatorics

35 IV. Ordered records: correlated random variables

36 Brownian positions Brownian records x 2 x 1 x 1 x 2 m 2 m 1 t t

37 First-passage kinetics: brownian positions Probability two Brownian particle do not meet t x 2 x 1 Universal probability S t = 2t t 2 t Sparre Andersen 53 Asymptotic behavior Feller 68 S t 1/2 Behavior holds for Levy flights, different mobilities, etc Universal first-passage exponent S. Redner, A guide to First-Passage Processes 2001

38 First-passage kinetics: brownian records Probability running records remain ordered x 1 x 2 m 2 m simulation t -1/4 t P t S t = ± Is ¼ exact? Is exponent universal?

39 m 1 >m 2 if and only if m 1 >x 2 x 1 x 2 m 2 m 1 t

40 From four variables to three Four variables: two positions, two records m 1 >x 1 and m 2 >x 2 The two records must always be ordered m 1 >m 2 Key observation: trailing record is irrelevant m 1 >m 2 if and only if m 1 >x 2 Three variables: two positions, one record m 1 >x 1 and m 1 >x 2

41 From three variables to two Introduce two distances from the record u = m 1 x 1 and v = m 1 x 2 Both distances undergo Brownian motion Boundary conditions: (i) absorption (ii) advection (u, v, t) t v=0 = 0 and u Probability records remain ordered P (t) = Z 1 Z 1 0 = Dr 2 (u, v, t) 0 v =0 u=0 du dv (u, v, t)

42 Diffusion in corner geometry v u

43 Backward evolution Study evolution as function of initial conditions P P (u 0,v 0,t) Obeys diffusion equation P (u 0,v 0,t) t = Dr 2 P (u 0,v 0,t) Boundary conditions: (i) absorption (ii) advection P v0 =0 = 0 and P u 0 + P v 0 =0 u 0 =0 Advection boundary condition is conjugate

44 Solution Use polar coordinates r = q u v2 0 and = arctan v 0 u 0 Laplace operator r 2 = 2 r r r r 2 2 Boundary conditions: (i) absorption (ii) advection P =0 = 0 and r P r P Dimensional analysis + power law + separable form P (r,,t) r 2 Dt f( ) = /2 =0

45 Selection of exponent Exponent related to eigenvalue of angular part of Laplacian f 00 ( )+(2 ) 2 f( ) =0 Absorbing boundary condition selects solution f( ) =sin(2 ) Advection boundary condition selects exponent tan ( ) =1 First-passage probability P t 1/4 EB & Krapivsky PRL 14

46 General diffusivities Particles have diffusion constants D1 and D2 (x 1,x 2 ) (bx 1, bx 2 ) with (bx 1, bx 2 )= x1 p D1, ben Avraham Leyvraz 88 x p 2 D2 Condition on records involves ratio of mobilities r D1 D 2 bm 1 > bm 2 Analysis straightforward to repeat r D1 D 2 tan ( ) =1 First-passage exponent: nonuniversal, mobility-dependent = 1 arctan r D2 D 1

47 Numerical verification simulation theory β D 1 /D 2 perfect agreement

48 Properties Depends on ratio of diffusion constants (D 1,D 2 ) D1 D 2 Bounds: involve one immobile particle (0) = 1 2 (1) =0 Rational for special values of diffusion constants (1/3) = 1/3 (1) = 1/4 (3) = 1/6 Duality: between fast chasing slow and slow chasing fast D1 D 2 + D2 D 1 = 1 2 Alternating kinetics: slow-fast-slow-fast

49 Multiple particles Probability n Brownian positions are perfectly ordered Records perfectly ordered P n t n n = m 1 >m 2 >m 3 > >m n In general, power-law decay S n t n n(n 1) 4 Fisher & Huse 88 n n n /2 2 1/4 1/ Uncorrelated variables provide an excellent approximation Suggests some record statistics can be robust

50 Recap IV First-passage kinetics of extremes in Brownian motion Problem reduces to diffusion in a two-dimensional corner with mixed boundary conditions First-passage exponent obtained analytically Exponent is continuously varying function of mobilities Relaxation is generally slower compared with positions Open questions: multiple particles, higher dimensions Why do uncorrelated variables represent an excellent approximation?

51 First-passage statistics of extreme values Survival probabilities decay as power law First-passage exponents are nontrivial Theoretical approach: differs from question to question Concepts of nonequilibrium statistical mechanics are powerful: scaling, correlations, large system-size limit Many, many open questions Ordered records as a data analysis tool

52 Publications 1. Scaling Exponents for Ordered Maxima, E. Ben-Naim and P.L. Krapivsky, Phys. Rev. E 92, (2015) 2. Slow Kinetics of Brownian Maxima, E. Ben-Naim and P.L. Krapivsky, Phys. Rev. Lett. 113, (2014) 3. Persistence of Random Walk Records, E. Ben-Naim and P.L. Krapivsky, J. Phys. A 47, (2014) 4. Scaling Exponent for Incremental Records, P.W. Miller and E. Ben-Naim, J. Stat. Mech. P10025 (2013) 5. Statistics of Superior Records, E. Ben-Naim and P.L. Krapivsky, Phys. Rev. E 88, (2013)