Tuesday, May 08, 2007

the joseph effect in run differential

this writer is only casually familiar with the work of benoit mandelbrot, the brilliant mathematician and author of the seminal "fractal geometry of nature" and "fractals and scaling in finance". but sometimes it takes only a casual familiarity to spur larger questions and examinations, and in this case that has again proven to be true -- this time in relation to baseball.

much thought in cubdom has recently been devoted to run differential, as the cubs were fashioning a very strong positive run differential in spite of a losing record that put the cub beyond important probability benchmarks. many took this to indicate an imminent turnaround for the better. although this page disagreed with the import placed upon and conclusions drawn from the team's run differential at such an early stage, the more important question persisted -- what exactly does run differential of recent games mean going forward?

a simple analysis would be simply to look back at several seasons and try to divine at what point in a normal season year-to-date run differential -- and the pythagorean estimated record that might be derived from it -- first approach the final figure. though perhaps inelegant, such an analysis is easy -- and the answer in examining the last four seasons (2003-6) suggests that run differential becomes material at a point approaching or in excess of 60 games, though even afterward the data can drift.

that is a point some time in the future still, and -- as can be seen by the variance of the 26-day moving averages of runs scored, runs allowed and run differential -- it remains difficult at this point in the season to divine much if anything useful from the performance of the club so far in these terms.

recalling mandelbrot, however, this writer was struck by some of the characteristics of these data sets.

fractal sets satisfy rigorous definitions, but the net effect in behavior of a fractal set is that they exhibit self-similarity, infinite variance syndrome (what mandelbrot called the noah effect, or the capacity to vary immensely and discontinuously from one event to the next) and long-range dependence (the joseph effect, or the measurable effect on the next event of past events of surprising age and distance).

a cursory examination of run differential shows that discontinuity from point to point is more than possible -- a sequence of two- and three-run games is commonly enough punctuated by a 15-run drubbing five times the magnitude of its surrounding examples with no warning. the existence of a noah effect seems assured.

but more interesting still was the idea that these sets may trend -- that is, that the past behavior may in fact help to forecast probable values of future data points -- that is again, long-range dependence.

before anything like prediction can be countenanced, the first hurdle to overcome is a demonstration of dependence that might indicate fractal properties. how might the data be shown to depend on past events?

this page undertook to chart the last four years of games for the chicago cubs by runs scored and allowed, with the aid of baseball reference, and look for dependence on large discontinuities -- that is, did run-related data exhibit different behavior following a blowout win or loss?

this writer chose to concentrate his examination on run differential. each 162-game year was examined in 152 overlapping ten-game segments, and the average single-game run differential for each segment was calculated and then averaged over all segments to give a baseline expectation.

then special conditions were imposed -- following a game in which the run differential was positive or negative over a certain number of runs, was there a measurable change? on the sample size of individual years, any such correlation was difficult to see.

2003following post W avg RD 10-span avg RD 10-span post L avg RD 10-span
4+ 0.066 0.116 0.262
 n=29 152 26
5+ 0.114 0.116 0.275
 21 152 20
6+ 0.086 0.116 0.479
 14 152 14
2004following post W avg RD 10-span avg RD 10-span post L avg RD 10-span
4+ 0.868 0.823 1.109
 40 152 22
5+ 1.000 0.823 1.350
 28 152 14
6+ 0.941 0.823 1.627
 17 152 11
2005following post W avg RD 10-span avg RD 10-span post L avg RD 10-span
4+ 0.152 -0.077 0.050
 29 152 30
5+ 0.222 -0.077 -0.233
 18 152 21
6+ 0.179 0.116 -0.500
 14 152 13
2006following post W avg RD 10-span avg RD 10-span post L avg RD 10-span
4+ -0.927 -0.873 -0.910
 30 152 42
5+ -0.911 -0.873 -1.124
 18 152 34
6+ -1.220 -0.873 -1.240
 10 152 25

it was in fact only in taking the more significant weighted averages of all four seasons together that the underlying view became clear.

2003-2006following post W avg RD 10-span avg RD 10-span post L avg RD 10-span
1+ 0.007 -0.003 -0.012
  304 608 304
2+ 0.055 -0.003 -0.020
  224 608 220
3+ 0.088 -0.003 -0.016
  176 608 170
4+ 0.103 -0.003 -0.046
 128 608 120
5+ 0.212 -0.003 -0.210
 85 608 89
6+ 0.136 -0.003 -0.205
 55 608 63

here, then, it becomes clear that over the last four seasons in the ten games following a three-run-or-more win, the cubs have recorded an average run differential of approximately 0.09 runs per game greater than usual -- that is to say, winning begets winning, and losing begets losing. just as importantly, the stronger the initial win or loss, the stronger its carryover effect -- as least to the limit observable here as sample size becomes an issue when discussing wins and losses decided by six runs or more.

the possible reasons for such a truth are manifold, and this page will leave speculation to the reader. but the fact of it is sufficient to suppose that not only is run differential discontinuous but also dependent -- it can, to at least some limited degree, be forecast based on immediately preceding events.

of course, to say that the data trend does not comprehend the whole of the picture -- if trends were never broken, a team once set upon winning would never cease to win, something we know not to be true. what remains, having established trend in run differential, is an examination of methods of forecasting -- can turning points in trend be articulated? such will be the basis of a subsequent post.

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