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Beyond the Asterisk * Adjusting for Performance Inflation In Beyond the asterisk * Adjusting for performance inflation in professional sports Alexander M. Petersen IMT Lucca Lucca, Italy Sunday, August 5, 2012 Bridging the past and the present 1. Method 3. Re-ranking for The All-Time “deflating” Greats achievement metrics 2. The Statistical Physics of Achievement Sunday, August 5, 2012 1. Establishing a baseline by removing trends 1394 YANHUI LIU et al. PRE 60 Financial Market Activity El Niño and La Nina courtesy of William S. Kessler, NOAA / Pacific Marine Environmental Laboratory Y. Liu, P. et al., The Statistical Properties of the Volatility of Price Fluctuations, Phys. Rev. E 60, 1390-1400 (1999). FIG. 8. ͑a͒ Semilog plot of the autocorrelation function of g(t). ͑b͒ Autocorrelation function of ͉g(t)͉ in the double log plot, with sampling time interval ⌬tϭ1 min. The autocorrelation function of A 8 D 10 1.2 WW II ! ! 1.7 ! WW Ig(t) decays exponentiallyCultureto zero within half an hour, C(t) " " ϳexp(Ϫt/␶) with ␶Ϸ4.0 min. A power law correlation C(t)ϳtϪ␥ FIG. 7. The 1-min4 interval intraday pattern for absolute1 price 12 ) 10 exists inThethe instrumental͉g(t)͉ for more rethancordthree goesdecades. back to1882.Note that Paleoboth evidence suggests f ! changes of the S&P 500Englishstock index ͑1984-1996!+ ! 2.0 ͒͑shifted͒ and for + Time series ( t ) 0.8 0 English (fiction) r graphs are truncatedthatat the El10first Niñoszero havalueve ofoccurC(t)r.eThed fosolidr millionsline in of years. ! P( f ) the absolute priceP( 10 changes, averaged for the chosen 500 companies ␥ English 1M ͑b͒ is the fit to the function 1/(1ϩt ) from which we obtainRock␥ & Roll ͑1994–1995͒. Thein sociallength ofandthe day is 390 minutes. (t) 0.6In order to Jazz English GB r -4 French > ! ϭ0.30Ϯ0.08. The horizontal dashed line indicates the noise level. avoid the detection10 of spuriousEnglish US correlations, this daily patternGermanis i 8 natural phenomena Russian polyphony removed. Otherwise one finds peaks in the power spectrum0.4 at the -10 -8 -6 -4 -2 Americanism frequenciesareof 1/day typically10 and10larger. non-10 Note10 that10 both the curves1850 have 1900a where1950 the index2000 j runs 6over all theRepatriationtrading days N in the year, t Antibiotics WWI WWII f (word frequency) 13-year period (Nϭ3309 in our study͒ and tday denotes the 6 similar patternB withstationary:7 large values within the first 15 min1.2 afterAmericanthe WW II (t) / <f 10 10 Civil War time in the day. In orderi to avoid the artificial correlation f 4 4 Chinese market opens. English 10 6 G. B. Vietnam 0.26(1)there are underlying Language 1 WWII Korea (t) 10 U. S. caused by this daily oscillation, we remove the intraday pat- 2 0.77(2) b0.51(1) = 0.51(1) 10 w All similar exponent5 ␮Ϸ3 for the cumulative distribution, with a WWI 0 tern from G(t) which we2 schematically write as N 10exogenous lipopeptides 10 ( t ) 0.8 2 4 6 8 6 8 10 r glycylcyclines Fluctuation scale, Fluctuation (t) drop-off at low values. English 10 10 10 10 10 10 10 ! penicillin+ oxazolidinones w and endogenous factors Eng. GB C " = 0.29(1) G. B. 0.6 Eng. US g͑t͒ϵG͑t ͒/A͑t ͒, ͑10͒ N 0 English 6 0 day day 106 100 French 0.29(1) 10 1M V. CORRELATIONS10 IN THE VOLATILITYU. S. 1850 1900 1950 2000 Fict. that can Englishsignificantly All Chinese 0.49(1) (t) 0.52(1) 0.4 year, t r 5 for all days. Each data point g(t), denotes the normalized ! A. M. Petersen, J. Tenenbaum, S. Havlin, H. E. Stanley. 10 64 5 7 6 8 7 9A.8 Volatility109 fluctuate7 correlations8 !8 9for S&P10 500 119stock index1850 1900 1950 2000 1010 1010 10 10 10 1010 1010 10 10 10 10 10 1010 6 year,absolute t price changeStatisticalat time Lawst, which Governingis computed Fluctuations byin Worddivid- Use from Word Birth to Word Death. Unlike price10 changes that are correlated only on very Scientific Reports 2, 313 (2012). 60 English 0 HebrewCorpus Size, Nu(t) Year,ing teach point G(tday) at time tday of the day by A(tday) for 1010 German 105 1M short time scales10 ͓40͔͑a few minutes0.47(1)͒, the absolute values of Fict. 0.18(1) 0.60(1) French all days. Sunday, August4 5, 20120.65(1)0.24(1) 5 10 10 price changes show long-range power-law correlations on Three methods—correlation function, power spectrum, 66 7 7 8 8 9 9 10 4 5 8 6 7 8 9 1010 1010 10 10 10 10 10 10 10 1010 10 1010 ( t ) time scales up to a year or more ͓20–31͔. Previous works and detrended fluctuation analysis ͑DFA͒— are employed to r 7 Vocabulary size, 10 6 ! 60 have shown that0 understanding the power-law correlations, quantify the correlation of the volatility. The pros and cons 1010 Russian 10 Spanish 5 German 0.39(1) 10 0.22(1)specifically the 5values of the exponents,0.22(1)0.51 can be helpful for of each method and the relations between them are described 4 0.65(1) 10 Hebrew 10 guiding the selection of models and mechanisms ͓32͔. There- in the Appendix. 6 5 76 7 8 8 99 65 7 6 8 7 9 8 10 10 1010 10fore,10 10in this1010 part10we focus1010on the10quantification10 10 10 of power-law 0 Corpus size,0 N (t) 10 correlations of10the volatility.u To quantify the correlations, we 2. Correlation quantification Russian 0.12(1)use ͉G(t)͉ instead Spanishof V (t), i.e., time window T is set to 1 T 0.24(1) Figure 8͑a͒ shows the autocorrelation function of the nor- 4 5 6 min7 with8 ⌬9 tϭ1 6min for7 the best8 resolution.9 10 10 10 10 10 10 10 10 10 10 malized price changes g(t), which shows exponential decay with a characteristic time of the order of 4 min. However, we Corpus size, N1.uIntraday(t | fc) pattern removal find that the autocorrelation function of ͉g(t)͉ has power law It is known that there exist intraday patterns of market decay, with long persistence up to several months, Fig. 8͑b͒. activity in the NYSE and the S&P 500 index ͓23–25,42͔.A This result is consistent with previous studies on several eco- possible explanation is that information gathers during the nomic time series ͓20–28,40͔. time of closure and hence traders are active near the opening More accurate results are obtained by the power spectrum hours, and many liquidity traders are active near the closing ͓Fig. 9͑a͔͒, which shows that the data fit not one but rather Ϫ␤1 hours ͓25͔. We find a similar intraday pattern in the absolute two separate power laws: for f Ͼ f ϫ , S( f )ϳ f , while for Ϫ␤2 price changes ͉G(t)͉ ͑Fig. 7͒. In order to quantify the corre- f Ͻ f ϫ , S( f )ϳ f , where lations in absolute price changes, it is important to remove this trend, lest there might be spurious correlations. The in- ␤1ϭ0.31Ϯ0.02, f Ͼ f ϫ , ͑11͒ traday pattern A(tday), where tday denotes the time in a day, is defined as the average of the absolute price change at time ␤2ϭ0.90Ϯ0.04, f Ͻ f ϫ , ͑12͒ t of the day for all days: Saturday, July 21, 12 day N and j ͚ ͉G ͑tday͉͒ jϭ1 1 A͑t ͒ϵ , ͑9͒ f ϭ minϪ1, ͑13͒ day N ϫ 570 Accounting for Inflation Just as the price of a candy bar has increased by a factor of ~ 20 over the last 100 years (roughly 3% inflation rate), Sunday, August 5, 2012 72 The European Physical Journal B A B 100 > 100 D 90 90 80 80 70 70 60 60 50 50 League average < K > 40 40 Detrended league average < K 1920 1940 1960 1980 2000 1920 1940 1960 1980 2000 Year Year CD14 14 > D 12 12 10 10 8 8 6 6 4 4 League average < HR > 2 2 Detrended league average < HR 1920 1940 1960 1980 2000 1920 1940 1960 1980 2000 Year Year Fig. 4. Acomparisonoftraditionalanddetrendedleagueaveragesdemonstratestheutilityofthedetrendingmethod.Annual per-player averages for (A) strikeouts (B) detrended strikeouts (for pitchers),Accounting (C) home run, for and (D)Inflation detrended home runs (for batters). The detrended home run average is remarkably constant over the 90-year “modern era” period 1920–2009, however there remains a negative trend in the detrended strikeout average. This residual trend in the strikeout average may result from the decreasing role of starters (resulting in shorter stints) and the increased role in the bullpen relievers, which affects the average number of opportunities obtained for players in a given season. This follows from the definition of the detrended average given by equation (10). A second detrending for average innings pitched per game might remove this residual trend demonstrated in Figure 5.Thesharpnegativefluctuationsin1981and1994–1995correspondtoplayerstrikesresultinginJust as the price season stoppage and a reduced average number of opportunities y(t) for these seasons. ! " of a candy bar has increased by a factor of ~ 20 removed (detrended) by normalizing accomplishments by where P is the average of P (t) over the entire period, " # over the last 100 the average prowess for a given season. years (roughly 0.04 2009 We first calculate the prowess Pi(t)ofanindividual 1 3% inflation rate), P Raising the moundP (’62)(t) . (8) player i as 0.035 ≡ 110 " # the home run t=1900 Pi(t) xi(t)/yi(t), (4) 0.03 " PED hitting ability ≡ <P(t)> era The choice0.025 of normalizing with respect to P ofis players arbi- has where x (t)isanindividual’stotalnumberofsuccessesout x 3 i trary, and we could just as well normalize with respectalso increased to by of his/her total number of opportunities y (t)inagiven 0.02 i P (2000), placing all values in terms of current “2000a significant US year t.
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