In a failed attempt to escape from statistics by reading a novel (Midnight in the Garden of Good and Evil, by John Berendt), I discovered a game called psycho dice. One of the main character, Jim Williams, explains it as follows.
“I believe in mind control,” he said. “I think you can influence events by mental concentration. I’ve invented a game called Psycho Dice. It’s very simple. You take four dice and call out four numbers between one and six–for example, a four, a three, and two sixes. Then you throw the dice, and if any of your numbers come up, you leave those dice stand-ing on the board. You continue to roll the remaining dice until all the dice are sitting on the board, showing your set of numbers. You’re eliminated if you roll three times in succession without getting any of the numbers you need. The object is to get all four numbers in the fewest rolls.”
Williams was sure he could improve the odds by sheer concentration. “Dice have six sides,” he said, “so you have a one-in-six chance of getting your number when you throw them. If you do any better than that, you beat the law of averages. Concentration definitely helps. That’s been proved. Back in the nineteen-thirties, Duke University did a study with a machine that could throw dice. First they had it throw dice when nobody was in the building, and the numbers came up strictly according to the law of averages. Then they put a man in the next room and had him concentrate on various numbers to see if that would beat the odds. It did. Then they put him in the same room, still concentrating, and the machine beat the odds again, by an even wider margin. When the man rolled the dice himself, using a cup, he did better still. When he finally rolled the dice with his bare hand, he did best of all.”
This is a follow-up of the post Power of running world records
As suggested by Andrew, plotting running world records could benefit from a change of variables. More exactly the use of different variables sheds light on a [now] well-known [to me] sports result provided in a 2000 Nature paper by Sandra Savaglio and Vincenzo Carbone (thanks Ken): the dependence between time and distance in log-log scale is not linear on the whole range of races, but piecewise linear. There is one break-point around time 2’33’’ (or equivalently distance around 1100 m). As mentioned in the article, this threshold corresponds to a physiological critical change in the athlete’s energy expenditure: in short races (less than 1000 m) the effort follows an anaerobic metabolism, whereas it switches to aerobic metabolism for middle and long distances (or longer…). Interestingly, the energy is more efficiently consumed in the second regime than in the first: the decay in speed slows down for endurance races.
The reason of this graphical/visual difference is simple. Denote distance, time and speed by D, T and S. I have plotted the log T~ log D relation, which gave with . When using the speed S as one of the variables, the relations are and with and to the first order because is close to 1. With Nature paper findings (with the opposite sign convention), the two s are (anaerobic) and (aerobic), ie and . My improper is indeed in between. The slope ratio is much larger (larger than 2) on a plot involving the speed, clearly showing the two regimes, than on my original plot (a few 10%), which is the reason why it appear almost linear (although afterthought, and with good goggles, two lines might have been detected).
Below is the S ~ log D relation (click to enlarge) on which it appears clearly that 100 m and 100 km races are two outliers. It takes time to erase the loss of time due to the start of the race (100 m and 200 m are run at the same speed…), whereas the 100 km suffers from a lack of interest among athletes.Achim Zeileis also provides an extended world records table and R code in his comment.
As an aside, Andrew and Cosma Shalizi also comment and resolve an ambiguity of mine: one usually speaks about power-laws without much precision of context, but there are mainly two separate sets of power-law models. Either power-law regressions, where you plot y~x for two different variables (this is the case here); or power-law distributions, ie the probability distribution of a single variable x is , or extensions of that (with lots of natural examples, ranging from the size of cities to the number of deaths in attacks in wars).
A quick post on a one-day seminar on Monte Carlo methods for inverse problems in image and signal processing, that will take place at Telecom ParisTech on Tuesday, November 15th. Details and abstracts are on the seminar’s webpage:
(for English-reading people, here is a google translated version). The seminar is organised by Gersende Fort, from Telecom and CNRS and the program looks very interesting, the topics are varied and fairly methodological. The webpage is in French but I think the talks are going to be in English, since there will be English-speaking people in the audience. I’m very happy to participate by presenting the Parallel Adaptive Wang Landau algorithm I’ve been blogging about lately, and Christian Robert is going to present our parallel Independent Metropolis-Hastings paper, so I can’t wait to getting more feedback on both.
See you on Tuesday?