Ichiro's Record Based on Daily Care

Seattle Mariners outfielder Ichiro Suzuki reached 200 hits for the ninth consecutive season on Sunday, September 13, 2009, with an infield single. Thus he broke the Major League Baseball (MLB) record owned by Willie Keeler since 1901. You can read more about Ichiro's accomplishment at the Web site of MLB [1].

One of Ichiro's teammates Kenji Jojima told a reporter about the secrets of Ichiro's successful batting as follows [2]:

For example, Ichiro's chair at the clubhouse in our home city is a common one made of pipes, though we have comfortable sofas. Ichiro says, "Sitting on a spongy chair for a long time gives me a stress on my waist." . . . As for the pathway from the clubhouse to the baseball ground, Ichiro has his own choice. There are steps and a slope, and he always uses the latter for going up and down. During the four years of my observing him, he has not changed this behavior. Walking on the steps has the possibility of slipping and causing a sprain when one has spike shoes. . . . On coming back to the bench after hitting a home run, Ichiro says to me, "Jo, did my back look like pleased?" I say, "Yes, it did." Hearing this, he says with a bitter smile, "Then I'm not yet good enough." Such a deed with emotion appearing outside the body as running joyfully is a bad thing for him. . . . (Translated from Japanese by the author.)

Ichro's wonderful record is based on such great daily care of his body and mind. I was much more impressed by Jojima's talk than by the news of Ichiro's breaking of the MLB record after 108 years.

References

1. "Chasing History: 9 Consecutive Seasons with 200 Hits," mlb.com.
2. K. Jojima, "Let's talk on Ichiro (1)," Asahi-shimbun (September 15, 2009).

Bolt's New 200-m Record and the Limiting Value

Fitting an exponential function to the data on men's 200 m world record progression.

On August 20, 2009, the Jamaican sprinter Usain Bolt set again a world record of 19.19 seconds in the 200-m sprint at the World Championships held in Berlin, adding to the gold he won in the 100 m.

I have taken the data on men's 200-m world record from Ref. 1, and have tried the least squares fit of an exponential function to the data. The best-fit curve obtained are shown in the above figure together with the data used. The result shows that the limiting value of the record would be 18.81 seconds.

The computation of the fit has also yielded the probable error of plus or minus 0.52 seconds for this value, though the error larger than plus 0.38 seconds cannot actually happen. This indicates that the accuracy of the prediction of the limiting value by such curve fitting is quite poor.

Reference

1. "Men's 200 metres world record progression," Wikipedia, the free encyclopedia (24 August 2009 at 01:02)

Last revised: August 26, 2009.

Bolt's World Record Changes Empirical Prediction Again

World record progression for men's 100 m. Data, from Ref. 1; dotted line, least-squares fit of exponential function to data through 2005; red line, fit to data through 2008; and purple line, fit to data through 2009. (You can see the real size image by clicking on the image).

Until the year of 2005, the plot of the world record for men's 100 m sprint as a function of year allowed a good fit by an exponential function with an additive constant, value of which meant the possible limiting value of the record. The data through 2005 gave the limiting value of 9.66 seconds. The data through 2008, however, predicted an improved limiting value of 9.43 seconds owing much to Usain Bolt's record, in the Beijing Olympic Games, of 9.69 seconds, which was one of outliers with respect to the fitted curve (Ref. 2).

Now his record of 9.58 seconds in Berlin on August 16, 2009, again changed the limiting value down to 9.09 seconds (with errors of plus or minus 0.49 seconds). When a single new record affects the possible limiting value this much, it would mean the following two facts: (1) The application of exponential fitting to data on men's 100 m sprint is completely inadequate. (2) Bolt's running ability is extraordinarily and exceptionally wonderful.

Notes added later:

Originally the title of this article was "Bolt's World Record Changes Statistical Prediction Again," but I replaced the word "Statistical" by "Empirical." The reason is this: The method used is often called statistical, but I think it better to be called empirical. Further, I'm not a statistician but a maker of empirical formulas.

A related blog article appeared: Michael Banks, Bolt out of the blue, physicsworld. com (August 17, 2009).

Read also Bolt's New 200-m Record and the Limiting Value.

1. "World record progression 100 metres men," Wikipedia, The Free Encyclopedia (18 August 2009 at 00:44).
2. "World Records for Men's 100 m Defy Simple Curve Fitting (2)" Femto-Essays (September 14, 2008).

Last revised: August 24, 2009.

World Records for Men's 100 m Defy Simple Curve Fitting (2)

World record progression for men's 100 m. Data, from Re. 4; dashed curve, least-squares fit of exponential function to data up to 2005; solid curve, least-squares fit of exponential function to data up to 2008. (You can see the real size image by clicking on the image).

The figure and some words in my previous blog article [1] were cited by articles of Wired Science [2] and other Web sites ([3], for example). Considering much interest shown to the topic, I post here a sequel to the previous article.

The data and the solid curve in the figure given above are the same as those in the previous article. As for another curve, description is made in the following paragraph. Following the example of the graph on a Wikipedia page [4], I have attached error bars to the earlier data recorded by hand timing (1912–1976). The later data were taken by electronic timing, and errors are considered to be within the size of the circle used for plotting. The curves fitted to the data pass through data points within the error bars for hand-timing days. (These were good old days!) Grids have also been drawn in the revised plot for the ease of reading off of values.

Alexis Madrigal writes in his article [2], "Though no statistician we spoke with had recalculated their numbers, the new world record is likely to rejigger the equations they use to calculate the maximum human speed." In relation to these words, the following is to be noted: The curve in my previous figure is a "recalculated one" in the sense that all the data including the one established by Usain Bolt at the Beijing Olympic Games are taken into account in the least-squares fit. To make this clearer, I have plotted another curve (shown by dashed line) obtained by a least-sqaures fit to the data up to 2005. This curve shows an asymptotic value of (9.66 ± 0.07) seconds.

The model of exponential decrease neglects the trends of small jigging of data, causing only changes in three coefficients in the equation (see Appendix 1). These coefficients are related to the height at the start of the curve, the rapidity of decrease and the asymptotic height of the curve. As was described in the previous article, the new asymptotic value of the world record was (9.43 ± 0.17) seconds.

In the previous article I compared this value with the value of 9.48 seconds given by Kevin Duffy in 2002 [5]. However, an examination of this value has shown that his curve is not the best fit to the data; using a logistic function and the data listed by him, I have obtained a better fit to the data with an asymptotic value of (9.67 ± 0.26) seconds, which is in good agreement to the value of 9.66 seconds obtained with data up to 2005.

Therefore, Duffy's curve fit should be regarded as the result of insufficient search with a fortuitously low asymptotic value, and the comparison with the new asymptotic value of 9.43 seconds should be made with 9.67 or 9.66 seconds. Though all these values agree with each other within the errors of the least-squares fit (see the error bars attached to the right ends of the curves), the decrease of the central value by 0.24 or 0.23 seconds obtained in the fit to the data up to 2008 reflects the big effect produced by Usain Bolt's two latest records. The larger error in the asymptotic value of the fit to the data up to 2008 indicates that the rapid decrease of the record time brought about by Bolt defies simple curve fitting.

I wrote in the previous article, "During many years, unexpected factors might come to affect the making of records, so that the result of curve fitting should not be much relied upon." The scientists whom Madrigal spoke with also said things similar to this. Namely, Peter Weyand, a physiologist at Southern Methodist University in Dallas who focuses on the biomechanics of running, said that mathematical models could never predict how fast humans might eventually run. The biomechanicist John Hutchinson of the Royal Veterinary College at the University of London, who studies how animals move, agreed with Weyand that the human speed limit would remain impossible to predict with any confidence, and mentioned as limiting factors the amount of advanced biotechnologies the International Olympic Committee and other regulatory authorities would allow sprinters to use.

In the previous article I criticized Duffy's use of a logistic function, but I do not deny the use of one when the earliest data with the trend of slow decrease are available. I expect that such data will soon be posted on the Wikipedia page [4], because it now includes the title of a section, "Unofficial progression before the IAAF" (IAAF was the International Amateur Athletics Federation, and now is known as the International Association of Athletics Federations).

1. "World records for men's 100 m defy simple curve fitting," IDEA & ISAAC: Femto-Essays (18 August 2008); also in Ted's Coffeehouse (19 August 2008).
2. A. Madrigal, "Bolt is freaky fast, but nowhere near human limits," Wired Science (25 August 2008).
3. Fabulation, "Usain Bolt : rapide, mais pas surhumain," Geek… mais pas trop (27 August 2008).
4. "World record progression 100 metres men," Wikipedia, The Free Encyclopedia (6 September 2008, at 05:52).
5. Kevin Duffy, 100 m sprinting: Is there a limit? (21 September 2002; last revision, 15 January 2008).

Read a similar analysis made on August 18, 2009, by taking Bolt's record of 9.58 seconds into account.

Appendix 1

The equation I used for curve fitting is as follows:

y = a + b exp(–cx),

where I used transformations

x = (X – 1900)/100,
y = Y – 9.5 (seconds),

X denotes the year, and Y, the world record in seconds. These transformations make the estimation of the starting value of the coefficients a, b and c for least-squares fit easy. A simple guess of the the starting values of a=0, b=1 and c=1 is good enough. The final values of coefficients obtained for the data up to 2008 are as follows:

a = –0.07 ± 0.17 (seconds),
b = 1.30 ± 0.13 (seconds),
c = 1.27 ± 0.36.

The asymptotic value of the world record is given by (9.5 + a) seconds.

Appendix 2

The author of the blog article [3] wrote me, "Some friends of mine asked me why you had chose an exponential model, and not a polynomial one or another mathematical function. I answered that in physics it is very usual to see things behave in an exponential way. However, it is true that sportsmen are neither radioactive nuclei nor physical objets that are inclined to have an exponential behavior. What do you think I could answer to this particular question?" In this appendix I write my answer to this question.

Sure, sportsmen are neither radioactive nuclei nor any physical object. However, the record produced by them shows the following trends: In earlier years of taking records, they can improve the record rather easily by efforts and exercise to get over the past records, the time of record exhibiting a rapid decrease. Then the stage of slow decrease comes because of limitation by mankind's bio-mechanical structure, showing the trend of approach to a limit. The simplest function to express these global trends is an exponential function with a constant term. A polynomial, for example, does not guarantee the reproduction of the trend of approach to a limit, though it can express minor jigging in data.

Ranking Countries in Order of Athletic Level

I never attach importance to the number of gold medals won by different countries in Olympic Games, nor want to find fault with that attained by China in Beijing Olympic Games held this summer. I would just like to point out that the ranks of countries are quite different when the number per population is considered instead of the actual number.

The ranks of countries by the number of gold medals (more than five) in Beijing Olympic Games are as follows (Ref. 1):

1	China	51
2	United States	36
3	Russian Fed.	23
4	Great Britain	19
5	Germany	16
6	Australia	14
7	Korea	13
8	Japan	9
9	Italy	8
10	France	7
10	Ukraine	7
10	Netherlands	7
13	Jamaica	6

On the other hand, the ranks by the number of gold medals per population among the above thirteen countries are as shown below. The first number for each country is the population taken from Ref. 2 in units of one hundred million; and the second number is the number of medals per population multiplied by one hundred million.

1	Jamaica	0.0271	221
2	Australia	0.214	65
3	Netherlands	0.164	43
4	Great Britain	0.610	31
5	Korea	0.482	27
6	Germany	0.822	19
7	Russian Fed.	1.42	16
8	Ukraine	0.461	15
9	Italy	0.596	13
10	United States	3.05	12
11	France	0.645	11
12	Japan	1.28	7
13	China	13.3	4

If we include the countries and regions that won at least one gold medal, the ranks would be different much more. Anyway, the second method of ranking clearly shows that a good fight exhibited by Jamaican athletes including the sprinter Usain Bolt was wonderful in Beijing Olympic Games.

Let us consider two countries A and B, A having much larger population than B, and assume that these countries have the same athletic level. Then, country A is statistically expected to produce a larger number of excellent athletes. This makes the method of ranking by the number of medals favorable to A. In country A, however, the selection of the delegation for Olympic Games would be severer because of the limited number of medals expected, making the method of ranking by the number of medals per population unfavorable to A. Therefore, the method that combines these two methods with certain weights might be adequate to know the proper rankings of countries in order of athletic level.

1. Overall Medal Standings, The Official Website of Beijing Olympic Games August 8-24, 2008.
2. List of countries by population, Wikipedia, the free encyclopedia (26 August 2008, at 07:09).

World Records for Men's 100 m Defy Simple Curve Fitting

World record progression for men's 100 m. Data, from Re. 1; curve, least-squares fit of exponential function to data. (You can see the real size image by clicking on the image).

Jamaican sprinter Usain Bolt won the gold medal for the 100-meter race of the 2008 Olympic Games on Saturday, August 16, establishing the new world record of 9.69 seconds. We find world records for this race since 1912 at the Wikipedia site [1]. A few data points of the latest world records show rapid decrease (see the figure above). This trend seems to defy simple curve fitting.

However, I dared to try fitting of an exponential function, y = a + b exp(−cx), to the data. The asymptotic value a, i.e., the limit of the world record, has been found to be 9.43 seconds with a probable error of 0.17 seconds, namely, to lie between 9.26 to 9.60 seconds. During many years, unexpected factors might come to affect the making of records, so that the result of curve fitting should not be much relied upon.

Duffy [2] also considered the limit to 100-m sprinting. He fitted a logistic function to the data up to 2002, and estimated a limit of 9.48. This value is rather in good agreement with the present result.*

A logistic function is useful to model the S-curve of growth [3]. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops. To use this function for the decay phenomenon that reaches a limit, it is necessary to make the function upside-down by making the coefficient in the exponential function negative. Further, one more coefficient should be added to give a finite limit. The function thus obtained has the properties of a slow initial decrease and a final decrease of approximately exponential type. To model a data set without a slow initial decrease, the exponential function of the form I used suffices.

* Originally I wrote here the following sentence put in parentheses: Among the legends in his Figure 2, "exponential fit" should read "straight-line fit". However, this was wrong. Duffy actually used the function of the form y = b exp(−cx) [4]. Reference 2 now includes new results (see sections of [2] entitled "The Performance Enhancing Drugs (PEDs)" and "100 Years - The Jamaicans"). (This note written August 17, 2012)

References

1. "World record progression 100 metres men," Wikipedia, The Free Encyclopedia (18 August 2008, at 12:42).
2. Kevin Duffy, 100 m sprinting: Is there a limit? (September 21, 2002; last revision, January 15, 2008).
3. "Logistic function," Wikipedia, The Free Encyclopedia (August 11, 2008, at 12:38).
4. Kevin Duffy, private communication (August 14, 2012)

Note added later:

• Read part 2 of this article.
• Read a similar analysis made on August 18, 2009, by taking Bolt's record of 9.58 seconds into account.