### Mathematical Regimen for Projecting Seasonal Football Wins

Historical Overview

If wins and losses were the perfect rating system for teams in the National Football League, the team with the most wins at the end of the year would win the Super Bowl. Of course, that rarely happens. This leads to the question that has been stumping analysts for many years: what *is* the best way to determine which teams have the highest probability of winning the league championship?

The process began some time in the 1960's and 1970's. People like Virgil Carter began to use the new computer technology to determine things like the value of field position. Then in the 1990's books began to be published, most notably "The Hidden Game of Football" which opened many eyes to the fact that the current rating systems and statistics were full of errors [1]. In the early 2000's Aaron Schatz and some of his college friends started a group called the Football Outsiders. They posted articles about new statistical research that has grown in popularity quite quickly.

Today there are hundreds of web sites and publications that claim to have the most accurate rating system. There are a variety of factors that are commonly used, such as points for and against, yards gained and allowed, and turnover rate just to name a few. Some systems rate individual players, some rate teams as a whole. Some systems such as Aaron Schatz' DVOA (Defense-Adjusted Value Over Average) have been able to incorporate nearly every variable that exists in football.

In the end, it seems that no matter what each system might use to rate teams, there is never going to be a perfect method. There are too many unquantifiable factors in the game, such as luck or how the ball bounces. The players themselves are mostly unpredictable as their mental and physical abilities fluctuate just as any human being's abilities would.

The daunting task of creating the most accurate rating system is far from being finished, but as analysts all over the world share their ideas and slowly unravel the mysteries behind the game, there is still hope that some day a perfect system will be discovered. The perfect system will not only correlate well to past and current wins and losses, but will correlate well to future wins and losses also.

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Logic Motivations

The application of the scientific method in sports statistics is in its infancy. In this atmosphere of fresh territory for science applications, a new approach of definitive scientific methodology must be applied. In order to begin, one must examine the logic background of the sport.

Logically, the team that wins the most games is the champion of the season, and the teams that win the most score the most points. As far as parameterization, the closing lead or loss "C" is to be compared to a team's highest lead "H" during the game and the team's lowest deficit "L" during the game. The high "H" is greater than or equal to zero, while the low "L" is less than or equal to zero. The close "C" will always fall in between "H" and "L", but can possibly be equal to either one as well. Therefore, the teams that have a closing value of "C" above zero will be winners, and comparing this "C" value to the "H" value with respect to the "L" value will produce a ratio for measurement; one may call this the Game Closing Factor or "GCF", or the Seasonal Closing Factor "SCF" over an entire season.

However, this logic breaks down when we consider the teams that win many games by a small margin, but have to make come-backs in order to do it; such teams have a smaller likelihood of winning per game. Although the most obvious factor affecting the modeling of a season is how high a lead "C" a team closes a game with, this still means that there must be a second factor involved. In fact, if one considers the case where a team has a close of zero every game, but had to come back from a large deficit to tie in each game, then the GCF will be at its maximum, meaning the team would have been calculated to win all the games of the season rather than the obvious one-half ratio. In this case, the closing factor alone is definitely insufficient.

To continue the logical argument, one must examine the stability of a team in play. This is the second factor, which will be referred to here as the Game Stability Factor or "GSF", or Seasonal Stability Factor "SSF" for an entire season. Logically speaking, the way in which a team wins should have an effect on their win probability: a team that wins by dominance will win more games than a team that wins by making grand come-backs from way behind. In this way, a team that has a zero closing average will be projected to win half of the season games regardless of whether the had a lead or a deficit. The team with the zero closing average over a lead will have a GCF of zero and a GSF of one since they remained in the lead the whole time. The team with the zero closing average over a deficit will have a GCF of one and a GSF of zero since they were behind the whole time. The graphic below in Figure 1 illustrates both the GCF and GSF effects in a visual representation.

Fig. 1

Figure 1 shows the relative positions of important values to the WTM, with bars showing the ranges that are compared in order to find such statistics as the GSF (first two bars, to the left) and the GCF (last two bars, to the right).

Since the logical arguments mandate that the GCF and GSF of a given team will, in equal parts, determine the basic GPW, it is immediately apparent that the equation;

1) GPW = ½(GCF + GSF)

is in force since the GCF and GSF are real numbers between the values of zero and one, and the GPW is a real number between the values of zero and one, essentially a probability of victory. Likewise, the SCF and SSF will follow the same relationship in determining the SPW, with the equation;

2) SPW = ½(SCF + SSF) **∙** n

in which the integer "n" appears, indicating the number of games in the season. While the SCF and SSF remain real numbers between the values of zero and one, the SPW then becomes a real number between the values of zero and "n", in essence the number of games won out of the total number of season games; this constitutes the Walters Trend Method.

Mathematical Theory

In a very real sense the SPW is telling us that, with all conditions being equal, the seasonal wins may be modeled from simple statistical calculations. Logically speaking, the SCF and SSF must be calculable for each team over the season. Going back to the logic arguments, one may assign a system of comparing the ranges, previously discussed, that produce the said SCF and SSF values.

The most basic calculation is that involving the GCF, in which the range from "C" to "L" is compared to the range from "H" to "L". In fact, the comparison equation is simply of the form;

3) GCF = (C − L) / (H − L)

and results in a value between zero and one. This is essentially a ratio of the net points gained over the low to the total in-game range of divergence.

Next, the GSF may be calculated. Finding the lowest value "L" for any team playing any game in the entire season, one may establish this particular value as "L_{min}"; furthermore, the highest value "H" for any team playing any game in the entire season may be labeled as "H_{max}". With these two values set, one may then proceed via the equation;

4) GSF = (H / 2 + L / 2 − L_{min} / 2) / (H_{max} / 2 − L_{min} / 2)

which measures the median of the range with respect to the seasonal maximum ranges of game scoring. The composite of the GCF and GSF gives the value of the GPW as outlined in eq. (1).

The GPW may then be averaged from game to game to produce the SPW. Also, the GCF and GSF may be averaged to produce the SCF and SSF, which also give the value of the SPW. The third method, which averages all values "L", "C", and "H" for the entire season, produces the SCF and SSF directly via the equations;

5) SCF = (**∑**^{n}_{i=1}(C_{i}) / n − **∑**^{n}_{i=1}(L_{i}) / n) / (**∑**^{n}_{i=1}(H_{i}) / n − **∑**^{n}_{i=1}(L_{i}) / n)

= (μ_{C} − μ_{L}) / (μ_{H} − μ_{L})

where "μ_{H}", "μ_{L}", and "μ_{C}" are the sample means (or average values) over the season of "H", "L", and "C" respectively [2]. This also requires that the SSF be constructed from the bottom up as well, so that;

6) SSF = (**∑**^{n}_{i=1}(H_{i}) / n + **∑**^{n}_{i=1}(L_{i}) / n − L_{min}) / (H_{max} − L_{min})

= (μ_{H} + μ_{L} − L_{min}) / (H_{max} − L_{min})

and the SPW follows from eq. (2). The determination of the SPW is the most important technical step of the WTM.

Technically speaking, the probability of winning "k" season games out of "n" total season games is defined by the binomial distribution;

7) P(k) = k! / ((n-k)! **∙** n!) **∙ **(SPW / n)^{k} (1 − SPW / n)^{n − k}

where "P(k)" is the probability of winning "k" games out of "n" total season games [3]. Furthermore, the probability of winning "i" games out of "k" season games (out of "n" total season games) becomes related to a binomial distribution;

8) SPW(k) = **∑ ^{k}_{i=1}**(i! / ((k-i)!

**∙**k!)

**∙**i (SPW / n)

^{i}(1 − SPW / n)

^{k − i})

where "SPW(k)" is the number of expected wins out of "k" games [4].

The error of actual seasonal wins "x_{i}" versus SPW_{i} for number of teams "m" may be measured by using the Pearson's chi-square test [5];

9) χ^{2} = **∑**^{m}_{i=1}((x_{i} − SPW_{i})^{2} / SPW_{i})

The chi-square value gives an error equal to the integrated probability;

*10) *ξ = ∫_{0}^{χ}^{2} (*z*^{15} / (2^{16} Γ(16)) **∙** *e*^{-z/2}) d*z*

with;

*11) *Γ(16) = ∫_{0}^{∞} *u*^{15} *e*^{-u} d*u*

where 0 ≤ *z* ≤ χ^{2} with 32 degrees of freedom [6]. Any trend with ξ ≤ 0.10 is inside the acceptable range for error, while any trend with ξ ≤ 0.01 is exceptionally precise.

The analytical phase of the WTM begins after the SPW's for all teams in all seasons have been determined. Trends should begin to appear as charts are created for the actual wins plotted against the SPW's for each season. Hopefully, cyclic oscillations will appear with a regular periodicity and amplitude that can also be further calculated and modeled for predictive purposes.

Method in Practice

The entire 2006 season has been analyzed using the Walters Trend Method. The correlation between the WTM estimated wins and the actual wins for each team were correlative at a value of .9108, with an r-squared value of .8296. Twenty-one of the thirty-two teams finished within 1.5 wins of their estimated total, as shown in figure 1 below. Only six teams deviated by two or more wins. The maximum deviation was 2.8 games, by two teams. The twelve teams that advanced to the post-season were all found within the top sixteen in WTM estimated wins.

Fig. 1

Figure 1 shows the correlation between actual games won and WTM-projected wins in a season for 32 teams in a standard 16-game season of the NFL. This particular season occurred in 2006.

When compared to the Football Outsiders' DVOA (Defense-Adjusted Value Over Average) and the Pythagorean Method (authored by Bill James), WTM correlates better than either to actual wins for the 2006 season.

Future Applications

The high/low/close aspect of WTM has opened the door for stock market-based candlestick charts to be created, which give a visual representation of how teams fluctuate within a season. Regression to the mean can clearly be seen when a team is analyzed on a week-to-week basis using these charts. As stated above, the intent is to identify visually these cyclic oscillations in the same way that stocks in the financial markets are tracked using technical analysis tools.

The predictive power of WTM should be no surprise. Bill James, founder of the school of sabermetrics, discovered that points in favor and points against could be used to estimate wins accurately. He also found that in general teams winning more than their estimate would decline the following year, and vice-versa. WTM, being based on some of the same principles, should work in the same fashion.

WTM is also applicable to other sports. Testing has been done on college basketball and professional baseball with results similar to those found in professional football.

Conclusion

The Walters Trend Method is only one piece of the puzzle, but hopefully it sheds light on the importance of point distribution and margin of victory. The best teams, according to WTM, are the ones that not only win by a large margin but also maintain that margin instead of letting go at the end of a game. So many teams are inclined to ease up and they go soft in the final quarter if they have a large lead, but those teams are opening themselves up for future disappointment. Consistency is key.

References

[1] Josh Levin, "Number Crunching: Why Doesn't Football Have A Bill James?",

http://www.slate.com/id/2092863, 12/19/2003

[2] J. A. Rice, __Mathematical Statistics and Data Analysis__, Duxbury Press: Belmont, CA (1995), 2^{nd} Edition pgs. 120, 138, 189

[3] J. A. Rice, __Mathematical Statistics and Data Analysis__, Duxbury Press: Belmont, CA (1995), 2^{nd} Edition pg. 36

[4] J. A. Rice, __Mathematical Statistics and Data Analysis__, Duxbury Press: Belmont, CA (1995), 2^{nd} Edition pg. 120

[5] J. A. Rice, __Mathematical Statistics and Data Analysis__, Duxbury Press: Belmont, CA (1995), 2^{nd} Edition pgs. 241-243, 313

[6] J. A. Rice, __Mathematical Statistics and Data Analysis__, Duxbury Press: Belmont, CA (1995), 2^{nd} Edition pgs. 59, 177, A8

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