Time-Series Forecasting

illustration of transient, recurrent, and absorbing states but makes only passing mention of the current use of a tie-breaker after the game score reaches 6–6. The description of judging methods in  gure skating still refers to rating compulsory exercises (school  gures), which are now defunct. Section 5.5 presents a model for estimating the probability that Bob Beamon’s world record for the long jump, set during the 1968 Olympics, would be broken before the year 2000—even though Mike Powell broke the record in 1991. Readers looking for statistical applications to sports will almost certainly be disappointed. The only applications based on real data occur in Chapter 5. American students will be confused by the frequent use of unfamiliar notations when referring to familiar concepts. For example, the expected value of a random variable is M6X7, the variance is D6X7, the sample mean is m x (and later Q mx ), the sample variance is e Dx , covariance is Kxy, and the Cartesian plane is xOy. Other phrases that might need additional translation are the “law of distribution of a random variable” (probability function), “statistical mathematical expectation” (sample mean), “ ducial probability” (con dence level), and the new area of “applied mathematical statistics. ” The  rst application in this section is a short discussion of interval estimation for a 100-meter dash time, based on hypothetical training run times. The normal distribution is never mentioned; instead, readers are instructed to calculate arg ê ü 441C5=25, where arg ê ü is “a function inverse to the Laplace function” (p. 58). Note that , not 1ƒ, is used as the con dence level (or  ducial probability). When the interval is  nally computed, the authors misinterpret it as having the stated probability of capturing the runner’s result in the next race, rather than capturing the runner’s average 100-meter dash time. The longest statistical application in Chapter 5 is the development of a simple linear regression model to predict the world record in the pole vault based on the year, using data from 1957 to 1987. The assumption that the progression of world records should follow a linear trend is a dubious one—a fact illustrated by a plot showing a fast rise in the early 1960s (corresponding to the introduction of  berglass poles), followed by a slightly concavedown increase typical of most world record progressions. The authors make no mention of this fact (despite showing 7 negative residuals, followed by 18 consecutive positive residuals in the plot) and never consider the time series nature of the data. One should not be surprised that the correlation r D 0966 is positive when the data are constrained so that Y (the world record) can only increase with X (the year). The data are further complicated by the performances of Sergei Bubka, who set new world pole vault records on 17 different occasions between 1984 and 1994—often increasing his previous record by just a single centimeter each time (possibly motivated by an incentive-based reward system). The model produced in the book “predicts” record heights of 6.16 m in 1990, 6.36m in 1995, and an increase of almost 4 cm per year. The world record of 6.14m was set in 1994 and has not increased since that time. Another application in this chapter shows (in excruciating detail) how to calculate the correlations in a multivariate setting for data on the records of the top weightlifters in 10 weight categories. Formulas for computing those sample correlations (p. 65) contain numerous typographical errors, including the substitution, in two different places, of the letter u for a subscript that is intended to be li. The conclusion makes judgments about the strength of the computed correlations (e.g., that r D ƒ045 means that the variables are “hardly correlated”) without justifying the basis for those conclusions. Although Mathematics and Sports may not be the best book from which to get ideas about statistical applications in sports, some of the other mathematical applications—particularly in areas related to operations research—are more intriguing. The authors have made a good effort to draw illustrations from many different sports, although at times the terminology may appear a bit novel (e.g., on p. 72 the  ve positions on a basketball team are listed as “center,” “forward,” “guard,” “left side,” and “right side”). They also demonstrate a wide variety of mathematical techniques with relatively simple (albeit somewhat contrived) examples. Would this be a good place to send mathematics students who are looking for ideas for sports-related projects? If they were interested in statistics, Bennett’s (1998) or issues of Chance magazine would be more suitable alternatives. For some of the other mathematical applications (e.g., Markov chains, judging  gure skating, player rating systems), a student might  nd something here to spark an interest in further investigation.