Records in a changing world Joachim Krug Institute of Theoretical - PowerPoint PPT Presentation

Records in a changing world Joachim Krug Institute of Theoretical Physics, University of Cologne • What are records, and why do we care? • Record-breaking temperatures and global warming • Correlations between record events • Records in random walks and the stock market with Gregor Wergen, Jasper Franke and Miro Bogner

The fascination of records

The fascination of records Most expensive transfer in the history of soccer

The fascination of records World’s tallest building from 1880-1884

The world’s tallest buildings over time 900 800 700 600 CN Tower height (m) 500 Empire State 400 Tour Eiffel 300 200 Strasbourg 100 1840 1860 1880 1900 1920 1940 1960 1980 2000 year

Temperature records

The 2010 summer heat wave http://www.spiegel.de/

The 2010 summer heat wave http:// limateprogress.org/2010/07/05/heat-wave-global-warming/

Temperature records in the USA http://www.u ar.edu/news/releases/2009/maxmin.jsp based on G.A. Meehl et al., Geophys. Res. Lett. 36 (2009) L23701

Daily temperature at Parque del Retiro, Madrid, on October 24 25 Daily maximum temperature [C] - October 24 Madrid, Parque del Retiro 20 15 10 Measurements Upper record value Lower record value 5 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Year (1920-2010) • 7 upper records and 6 lower records in 90 years • How many records should we expect if the climate did not change?

Mathematical theory of records I N. Glick, Am. Math. Mon. 85 , 2 (1978) • A record is an entry in a sequence of random variables (RV’s) X n which is larger (upper record) or smaller (lower records) than all previous entries • Example: 1000 independent Gaussian random variables ⇒ 7 upper records

Mathematical theory of records II • If the RV’s are independent and identically distributed (i.i.d.), the probability n = 1 / n by symmetry for a record at time n is P • The expected number of records up to time n is therefore n 1 ∑ k = ln ( n )+ γ + O ( 1 / n ) � R n � = k = 1 where γ ≈ 0 . 5772156649 .... is the Euler-Mascheroni constant, • Because record events are independent, the variance of the number of records is = ln ( n )+ γ − π 2 n � 1 k − 1 � ∑ � ( R n −� R n � ) 2 � = 6 + O ( 1 / n ) k 2 k = 1 • In a constant climate we expect 5 records in 90 years and only 7.5 records in 1000 years!

Record-breaking temperatures and global warming R.E. Benestad (2003); S. Redner & M.R. Petersen (2006) • Question: Does global warming significantly increase the occurrence of record-breaking high daily temperatures? • Model: The temperature on a given calendar day of the year is an independent Gaussian RV with constant standard deviation σ and a mean that increases at speed v • Typical values: v ≈ 0 . 03 o C/yr, σ ≈ 3 . 5 o C ⇒ v / σ ≪ 1

The linear drift model R. Ballerini & S. Resnick (1985); J. Franke, G. Wergen, JK, JSTAT (2010) P10013 • General setting: Time series X n = Y n + vn with i.i.d. RV’s Y n and v > 0 • For large n the record probability approaches a finite limit lim n → ∞ P n ( v ) > 0

Approximate calculation of the record rate for small drift • Let Y n have probability density p ( y ) and probability distribution function � x dy p ( y ) . Then q ( x ) = n − 1 n − 1 � � ∏ ∏ P n ( v ) = dx n p ( x n − vn ) q ( x n − vk ) = dx p ( x ) q ( x + vk ) k = 1 k = 1 • For small v we have q ( x + vk ) ≈ q ( x )+ vkp ( x ) dx p ( x ) q ( x ) n − 1 + vn ( n − 1 ) dx p ( x ) 2 q ( x ) n − 2 = 1 � � ⇒ P n + vI n n ≈ 2 � dx p ( x ) 2 q ( x ) n − 2 with I n = n ( n − 1 ) 2 • For the Gaussian distributon a saddle point approximation for large n yields 2 √ π n ( v ) ≈ 1 n + v ln ( n 2 / 8 π ) P � σ e 2

Comparison to temperature data G. Wergen, JK, EPL 92 30008 (2010); G. Wergen, A. Hense, JK, arXiv:1210.5416

Data sets European data • Daily temperatures from 187 stations over 30 year period 1976-2005 • Constant warming rate v ≈ 0 . 047 ± 0 . 003 o C/yr, standard deviation σ ≈ 3 . 4 ± 0 . 3 o C ⇒ v / σ ≈ 0 . 014 • Gridded re-analysis data from the E-OBS project for 1950-2010 yields v / σ ≈ 0 . 012 for 30 year period 1980-2010 American data • Daily temperatures from 207 stations over 30 year period 1976-2005 • Lower warming rate and larger variability due to continental climate: σ = 4 . 9 ± 0 . 1 o C, v = 0 . 025 ± 0 . 002 o C/yr ⇒ v / σ ≈ 0 . 005 • Monthly data from 1217 stations over 50 year period 1960-2010 display similar trend but lower variability ⇒ v / σ ≈ 0 . 010

European data 1976-2005: Mean daily maximum temperature Full line: Sliding 3-year average

No trend in the standard deviation

Temperature fluctuations are Gaussian

Record frequency in Europe: 1976-2005 • Expected number of records in stationary climate: 365 30 ≈ 12 • Observed record rate is increased by about 40 % ⇒ 5 additional records

Mean record number: 1976-2005

Correlations between record events

Records from broadening distributions JK, JSTAT (2007) P07001 • RV’s X n drawn from p n ( x ) = n − α f ( x / n α ) with α > 0 • Simulations indicate sub-Poissonian fluctuations in the number of records, indicating that record events repel each other Example: Uniform distribution

Record correlations in the linear drift model G. Wergen, J. Franke, JK, J. Stat. Phys. 144 (2011) 1206 • Consider the quantity l N , N − 1 ( v ) = P N , N − 1 with P N , N − 1 = Prob[ X N record and X N − 1 record ] P N P N − 1 • l N , N − 1 ( 0 ) = 1 and l N , N − 1 ( v ) ≡ 1 for Gumbel-distributed i.i.d part • lim N → ∞ l N , N − 1 ( v ) exists for v > 0 but not necessarily for v < 0 • Small v expansion yields l N , N − 1 ( v ) ≈ 1 + vJ N ( v ) with � 2 J N ≈ − 1 2 N 4 d � − 2 NI N ≈ κ NI N N 2 I N dN where κ is the extreme value index of p ( x ) ∼ ( 1 + κ x ) − κ + 1 κ

• Records cluster (repel) for distributions broader (more narrow) than an exponential: 3.5 Pareto ( µ =2): c = 0.05 Pareto ( µ =3): c = 0.05 3 Pareto ( µ =4): c = 0.05 Str. Exp. (0.5): c = 0.05 Exponential: c = 0.05 Correlations l N,N-1 (c) Gaussian: c = 0.05 2.5 Uniform: c = 0.05 1 - no drift (c=0) 2 1.5 1 0.5 1 10 100 1000 10000 100000 Number of events N

A record-based test for heavy tailed distributions J. Franke, G. Wergen, JK, PRL 108 (2012) 064101 • Identifying heavy tailed distributions from small data sets is generally problematic Clauset, Shalizi, Newman, SIAM Rev. 51 661 (2009) • For a set of N variables expected to be i.i.d, choose many random subsets of size n < N • Add a linear trend to the data in each subset and compute the normalized joint record probability l n , n − 1 as a function of the drift strength v • Then l n , n − 1 ( v ) > 1 implies strong evidence for heavy tailed behavior • The test is non-parametric and robust against the removal of outliers • Fluctuations caused by the smallness of the original data set can be reduced (to some extent) by the combinatorial proliferation of subsets

Example 1: Gaussian vs. Lévy RV’s for N = 64 , n = 16 Gaussian, σ =1.0 1.15 Levy-stable, µ =1.3 From independent Gaussian RVs From independent Levy RVs 1.1 1.05 1 l 16, 15 (c) 1 0.95 0.8 0.9 0.6 F(x) 0.4 0.85 0.2 0 0.8 -2 0 2 x 0.75 0 0.5 1 1.5 2 2.5 3 c

Example 2: Subsets of size N = 64 from ISI citation data source: S. Redner, Eur. Phys. J. B 4 131 (1998) 2.4 2.7 2.6 2.2 2.5 l n, n-1 (c*) 2.4 2.3 2 2.2 2.1 2 1.9 1.8 1.8 l 16,15 (c) 12 14 16 18 20 22 1.6 n 1.4 dataset 1 1.2 dataset 2 dataset 3 1 0.8 0 2 4 6 8 10 12 14 16 c

Record correlations induced by rounding effects G. Wergen, D. Volovik, S. Redner, JK, PRL 109 , 164102 (2012) 18 ∆ =1 ∆ =2 16 ∆ =4 Analyt. ∆ =1 14 Analyt. ∆ =2 Analyt. ∆ =4 12 10 n R ∆ 8 6 4 2 0 1 1e+10 1e+20 1e+30 1e+40 1e+50 1e+60 n Number of strong records from rounded i.i.d. Gaussian RV’s

Random walks & market fluctuations

Records in random walks 30 25 20 15 X 10 5 0 -5 0 100 200 300 400 500 600 700 800 900 1000 n ⇒ 65 records in 1000 time steps

Records in random walks S.N. Majumdar & R.M. Ziff, PRL 101 , 050601 (2008) • Simple one-dimensional random walk is defined by n ∑ η k X n = k = 1 with i.i.d. RV’s η k drawn from a symmetric, continuous distribution φ ( η ) • Based on a theorem of Sparre Andersen (1954), the probability of having m records in n steps is found to be � 2 n − m + 1 1 � 2 − 2 n + m − 1 → √ π n exp [ − m 2 / 4 n ] P ( m , n ) = n 4 n / π ≫ ln n + γ • Mean number of records: � R n � ≈ � • This result does not require φ ( η ) to have finite variance ⇒ valid also for superdiffusive (Lévy) walks!