Calibration of predictions in sports

One of the most important tasks of forecasting sporting achievements
It is to increase the accuracy of calculations. As is known, the accuracy of the forecast depends on the length of time series, and the period of retrospection period anticipation, forecasting method, and other factors [2]. General opinion on how these factors affect the efficiency of higher prediction results, no. For example, some scientists believe that the longer the period of history (20-30 years), the more reliable the forecast. Others believe that the outlook for the future is better to include the next two to three years, and therefore forecasting time series does not need a long duration. Third
They find it impossible to describe the long time series of a single equation, and assume expanded into separate simpler ...

1. Introduction 3
1. The essence of forecasting 4
1.1.The concept of forecasting 4
1.2. Forecasting Methods 5
1.3. Factor as a predictor of sport achievement 7
2 Description of the three methods of forecasting 9
2.1 Market Forecast 9
2.2 Tipsters 11
3 Description of the Data 13
3.1 Data Set 13
3.2 Calculations of the Prediction Market Forecasts 14
3.3 Calculations of the Betting Market Forecasts 18
4 Forecast Accuracy of Three Methods 20
4.1 Evaluation Criteria 20
4.2 Forecast Accuracy of Each Method 21
4.3 Forecast Accuracy of Combinations of the Methods 23
4.3.1 Accuracy of Weighting-Based Combined Forecasts 23
4.3.2 Accuracy of Rule-Based Combined Forecasts 24
5. Results 26
5.1 Football 26
5.2 Baseball 31
Conclusions 38
References 40

Sporting achievements, which will in the near or distant future in a particular sport, always interested scientists, coaches and athletes.
This is understandable because of what will be sporting achievements in the sport after 2, 4, 8, 12 or more years, largely depends on strategy and tactics for the selection and training of future contenders for the world and Olympic medals.
For example, to prepare a high jumper, having a chance to medal in 2012, must be at least roughly to know as early as 2005, the level at which the fight will take place at these competitions. If among the jumper will win medals athletes who have overcome the bar at a height of 2,55-2,60 and the average age of the champions and finalists of the Olympic Games in the sport is 1921, then in 2005 to take stock of 13-14 -year-old capable of jumpers and prepare them to conquer this height..
Look to the future record of achievements possible with the help of sports prognostics. This is one of the very interesting and promising trends in sports science, which especially in the last decade developing intensively in many countries around the world.
At present, the major institutions are special research units (laboratories, sectors), whose responsibilities include the development of forecasts in the field of physical culture and sports. The object of study - elite sport.
The object of study- Problems of forecasting the highest achievements in sport.
Subject of research: To study the forecasting of of the results in the sphere of sports
Tasks:
1. Expand the concept of forecasting
2. To study the methods of forecasting
3. To consider the factors predicting higher sporting achievements

In the 1999-2000 and 2000-2001 seasons, round tournament had the following structure: two games on Friday, five games Saturday and two games on Sunday. In the season of 2001-2002, round tournament, not meant seven games on Saturday and two games on Sunday. Unlike many other sports, especially in the United States, the game of football has three possible outcomes: win, lose or draw. In case of a tie, each team receives one league point; in the case of a win, the winning team gets three points in the championship, and the loser none.For each round of the tournament, we have gathered the results of the game, the price of shares on the PM (www.bundesligaboerse.de) to predict the results of these games Tipster prediction (win, draw 7, lose) the most popular German sports magazine (Sport Bild)and fixed rates the chances of the largest German state bookmaker (Oddset). PM provided forecasts for 91.18% of all games (NPM = 837 games), so we collect Betting Odds for the same sample of games (NBET = 837). However, we have less observational predictions bugs "because sports magazine did not publish forecasts for the two games on Friday during 1999-2000 and 2000-2001, seasons. In addition, the magazine arbitrarily ignored the predictions for the games in a few weeks, It leaves us with NTIP = 721 predictions bugs.Thus, the predictions associated with all three methods are available for 678 games. While the number of observations is smaller than those of studies that analyze the forecast accuracy of betting markets (e.g., Pope & Peel (1989): 1,291 matches, Cain et al. (2000): 2,855 matches, Dixon & Pope (2004): 6,629 matches, Vlastakis et al. (2007): 12,841 matches, Graham & Stott (2008): 11,000 matches), it is substantially larger than those of studies analyzing the forecast accuracy of prediction markets (e.g., Jank & Foutz (2007): 262, Pennock et al. (2001): 161, Servan-Schreiber et al. (2004): 208, Spann & Skiera (2003): 152). Table 1 provides descriptive statistics regarding the number of observations and proportion of actual home victories, draws and away victories in each sample and season.These results correspond roughly to the English football league results; in Goddard & Asimakopoulos (2004) report, home team wins 45.3% of games away victories are found in 28.0% and attracts occur in 26.7% of all games.3.2 Calculations of the Prediction Market ForecastsThe PM we investigate attracted about 10 000 registered users in all, an average of 1,500 active participants at each round of the tournament. It is usually open on Thursday at 6.00 pm, and trading ended five hours later, at 11:00 pm 8 Thursday.On Friday, Saturday and Sunday, the football market remained open for five hours every day during the games, and then shut down every round of the tournament at the end of the last game on Sunday. The payoff function of each part of virtual shares depends on the number of points of league football team gains one tournament round. In the season of 1999-2000, eventually winning stake losing team is $ 100; Teams from the living room $ 200, and the winning team is $ 400.The minimum $100 payoff for a loss serves to avoid "penny stocks":1.The payoff-rule changed for later seasons. Each share of stock of a losing team was $0, of a drawing team $1 and of the winner was $3.where d Home Away g r s: cash dividend of a share of stock that models the number of league points the home (away) team gains in the gth game in the rth tournament round of the sth season, Z Home Away g r s: number of league points the home (away) team gains in the gth game in the rth tournament round of the sth season, Draw Win Loss (/ )ds Home Away g r s: cash dividend of a share of stock in the case of a draw (win/loss) in the sth season, Gr,s: index set of games in the rth tournament round of sth season, Rs: index set of tournament rounds of sth season, and S: index set of seasons.In each tournament round of the 1999–2000 season, all participants of the Soccer Market start with the same assets: 1,000 shares of each type of team stock and $500,000 (virtual), with the possibility of a maximum virtual loan of $500,000 at a 1% weekly interest rate. For the 2000– 2001 and 2001–2002 seasons, the endowment in each tournament round consisted of 1,000 shares of each type of team stock and $5,000 (virtual) cash, with no loans possible. Participants are treated the same, regardless of when they enter the football market. A participant may sell shares in accordance with his or her assessment of the results of the game through the sale of shares allegedly overvalued shares of the team, or the purchase of shares allegedly undervalued stock team. Portfolio values ​​from one tournament round do not carry over to the next round of the tournament; Instead, the stimulus includes a monetary reward for each round.At the end of each round of the tournament, the participant with the highest value (virtual) portfolio gets $ 150 (real) person with the second highest value gets $ 100 and third grade participant gets $ 50.There is no risk of actual financial loss. Table 2 follows the recommendations of Spann & Skiera (2003) to describe the design of the PM (table 2).Table 2: Design of the prediction marketTo determine outcome predictions from the PM, we use the stock prices of the team stocks at the end of trading on the first day, that is, the earliest possible end-of-trading point before the first game of a tournament round to predict all games of that round. Equation (3) represents the expected league points of a team in a specific tournament round in the 1999–2000 season; Equation (4) describes the 2000–2001 and 2001–2002 seasons according to the current stock price:where: Z PMHome Away g r s t : expected gain of league points according to the PM for the home (away) team at the tth point in time in the gth game in the rth tournament round of the sth season, P Home Away g r s t : price of a share of the home (away) team’s stock at the tth point in time in the gth game in the rth tournament round of the sth season, and T: point of time at the end of the game of the home (away) team in the gth game in the rth tournament round of the sth season. Predictions for game outcomes reflect the differences in the stock prices of two competing teams; we predict a win for the team with the higher stock price. We predict a draw as the game outcome only when the two competing teams achieve identical stock prices. After determining the prices of the home and away teams in a specific game and given that all outcome probabilities sum to 1, we can calculate the specific outcome probabilities3.3 Calculations of the Betting Market ForecastsWe use the fixed betting odds of the largest German state-owned bookmaker (Oddset), which employs decimal odds and charges a fee of 25%, included in the odds.3 That fee is substantially higher than the average margin of approximately 12% in most European (non–stateowned) bookmakers (Vlastakis et al., 2007) or the 5% in person-to-person betting on betting exchanges such as Betfair (Smith et al., 2006). We derive the bookmaker's forecasts from the betting odds by retrieving the implied probability of the different game outcomes and standardizing the probabilities to 1:WhereB Draw Home Away grs: standardized probability derived from betting odds of a draw (home team win/away team win) in the gth game in the rth tournament round of the sth season, u Draw Home Away grs: unstandardized probability derived from betting odds of a draw (home team win/away team win) in the gth game in the rth tournament round of the sth season, andq Draw Home Away grs: betting odds of a draw (home team win/away team win) in the gth game in the rth tournament round of the sth season.Therefore, if the decimal odds of a home win, draw and away win are, respectively, 1.7, 2.8 and 3.3, the standardized probabilities are 47.1%, 28.6% and 24.3%. The highest probability determines the forecast for the game outcome. Our results show that the bookmaker never assigns a draw with the highest probability. Furthermore, we calculate the expected gain of league points by the home and away teams in a game on the basis of the standardized probabilities for each of the three possible game outcomes:In Table 3, we display the shares of outcomes predicted by each method for each season and all three seasons together.4 Forecast Accuracy of Three Methods4.1 Evaluation CriteriaOur criteria for evaluating and comparing the three forecasting methods are as follows: 1. We calculate the percentage of hits for each method, that is, the number of correctly predicted games relative to the total number of predicted games. 2. We calculate the root mean squared error (RMSE) for the deviation between the expected and actual gains of league points for each of the two teams in every game (N: total number of games in sample):We calculate the amount of money the predictions of each forecasting method would have won on the betting market for three possible fee scenarios: (a) with the 25% fee of the (state-owned) betting company, (b) with a fee of 12%, which is common for most European (non–state-owned) betting companies and (c) with no fee. The calculated profit in all three scenarios indicates the value of each forecasting method. Specifically, the winnings without a fee (0%) show whether forecasting performance is better than the betting odds. The amount after subtracting the betting company's margin denotes whether punters can use the information to make money in a real-world betting market. The 12% fee reveals whether punters could earn money in a (competitive betting market) situation with a fee below the monopolistic fee (25%) of the state-owned betting market in Germany.In addition, we compare the forecasts of the three methods with those of a naïve model and a pure random draw model. The naïve model always predicts a home win, which is the most frequent game outcome (i.e., the naïve model is not strictly naïve, because it uses this information; Forrest & Simmons, 2000). The pure random draw model randomly predicts one of the three events with overall probabilities, which provides a forecasting accuracy of (h2 + d2 + a 2 ), in which h, d and a are the proportions of home victories, away victories and draws in our data set (Forrest & Simmons, 2000, p. 321). 4.2 Forecast Accuracy of Each Method In Table 4, we compare the hit rates of the PM, the naïve model, random picks and betting odds for the whole sample of 837 games. The PM yields a hit rate of 52.69%, greater than the total number of home victories (50.42%) and pure random picks (37.73%). Betting odds have a slightly higher hit rate of 52.93% and a slightly lower RMSE, but lead to lower profits. Differences between the PM and betting odds are insignificant, indicating that the forecast accuracy is comparable. Both methods outperform the naïve model of home wins.Table 4: Comparison of forecasting accuracy of prediction markets and betting oddsa) Percentage of improvement of PM over alternative method = [hit rate PM – hit rate of alternative method]/hit rate of alternative method (one-tailed binomial test for difference of hit rate of PM). b) Root mean squared error for the deviation between the expected and actual gain of league points for both teams in every game. The naïve model only provides an outcome prediction (home win), from which we derive the expected gain in league points. Thus, the comparability of the RMSE of the naïve model is limited to the RMSE of PM and betting odds predictions, which provide separate predictions for the expected gain of league points for each team. c) Profit measured as the (relative) return on betting.Table 5 displays the hit rates of the PM, betting odds and tipster for the overlapping sample of 678 games. At this time, PM provides a higher rate of hit and profits than betting odds, but also a higher RMSE. Again, the differences between the PM and the betting odds are not essential, and both methods are far superior bugs and a naive model. Forecasts beetle are particularly poor; even naive model is clearly superior to them. Thus, the accuracy of prediction of PM and compare betting odds and much better than a bug or a naive model.Table 5: Comparison of forecast accuracy of different methodsa) The sports journal did not predict all games; therefore, the comparison with the PM and betting odds depends on the same selection of games. b) Percentage of improvement of PM over alternative method = [hit rate PM – hit rate of alternative method]/hit rate of alternative method (one-tailed binomial test for difference to hit rate of PM). c) Root mean squared error for the deviation between the expected and actual gain of league points for both teams in every game. However, the tipster and naïve model only provide an outcome prediction (home win, draw or away win), from which we derive the expected gain in league points. Thus, the comparability of the RMSE of the tipster and naïve model is limited to the RMSE of PM and betting odds predictions, which provide separate predictions for the expected gain of league points for each team. d) Profit measured as the (relative) return on betting.These results are consistent with the predictions of correlation (Table 6), for which we encode forecast home victory as "1", draw as "0" and the victory as "-1". The correlation between the predictions of the PM and the bug is 0.436; between the PM and the naive model 0.216. Therefore, the prediction of these methods differ considerably. In contrast, the PM forecasts and betting correlation coefficient of .844, indicating that they are relatively close resemblance.However, the forecasts are far from being equal, which indicates that we might be able to exploit these differences by combining the results of the different methods. Table 6: Correlation between forecasts of different forecasting methodsThe sports journal did not predict all games; the comparison with PM and betting odds is based on the same selection of games.4.3 Forecast Accuracy of Combinations of the MethodsSeveral studies show that combining the results of different forecasting methods can improve forecasting accuracy (Armstrong, 2001; Batchelor & Dua, 1995; Blattberg & Hoch, 1990). Therefore, we test the forecast accuracy of a weighting-based combination of forecasts (Blattberg & Hoch, 1990), as well as various rule-based combinations of forecasts.4.3.1 Accuracy of Weighting-Based Combined Forecasts We follow Blattberg & Hoch (1990), who suggest a 50:50 weighting, thus a simply averaging of forecasts in a different setting. Therefore, we averaged the predicted number of league points for the home and away teams from the PM and betting odds (see Equation (11)). We exclude the tipster, which does not provide a forecast for the expected league points and offers fairly poor predictions.where: Z Home Away g r s : forecast of the weighting-based combination (PM/betting odds) for the expected league points of the home (away) team of the gth game in the rth tournament round of the sth season. We use the differences in the expected league points to predict a win for the team with more expected league points; we predict a draw when the teams have identical expected league points. This weighting-based combination establishes a forecast accuracy of 52.69% (N = 837), equal to the hit rate of the forecasts of the PM for all 837 games. It also yields profits on betting markets with 25%, 12% and 0% fees of –13.12%, –0.59% and 11.47%, respectively. Neither the hit rate (one-tailed binomial test, p > .5) nor the profits (two-tailed paired t-tests, p > .6) differ significantly from the forecast of the PM or the betting odds for the same sample of 837 games (compare Table 4 with Table 7). However, the RMSE of the weighting-based combination is lower than that of the PM, though higher than that of the betting odds. The results for the sample of the 678 games are very similar (compare Table 5, second row, with Table 7, last column): Neither the hit rate (one-tailed binomial test, p > .4) nor the profits (two-tailed paired t-test, p > .3) lead to significantly different results. Therefore, we conclude that our weighting-based combination of forecasts does improve the forecasts of the PM or the betting odds notably. 16 4.3.2 Accuracy of Rule-Based Combined Forecasts Thus far, we have forecast all games, but we might improve forecast accuracy by concentrating on selected games. This situation more accurately reflects the real world; punters can usually deliberately bet on only a selected number of games. Therefore, we analyze the quality of the following rules to select the games that we want to forecast: 1. Only forecast if the forecasts of PM and betting odds are the same. 2. Only forecast if the forecasts of PM and the tipster are the same. 3. Only forecast if the forecasts of betting odds and the tipster are the same. 4. Only forecast if the forecasts of PM, betting odds and the tipster are the same. Table 7 shows the results. The rule-based forecasts select between 380 (56.0%) and 778 (93.0%) games in each sample and increase the hit rate to 53.98% (rule 1), 56.85% (rule 2), 56.52% (rule 3) and 57.11% (rule 4). Thus, based on the rules of the combined forecasts increase hit rate, but none hit the improved rate is not significantly different (one-sided binomial test) from entering the PM speed, that is 54.28% for a sample of 678 games (table 5). Rule 4 reaches the highest hit rate (57.11%), but chooses the least number of games. Betting $ 100 on each game will result in a gain of $ 5,267 (13.42%), if the company's rates will not charge you.This amount is far less than those realized for the rules that choose more games, and much less than the total profit of weight based on the combined forecasts. Total profits are highest for PM forecasts ($ 10295 for all the games 837, $ 10984 for overlapping 678 games). This total profit more than achieved by weighting on the basis of the combined forecasts or rely on the company's bet ($ 9977 for all the games 837, $ 9146 for overlapping 678 games). Therefore, this result confirms the high accuracy of the forecasts of IMS and betting odds.Table 7: Rule-based and weighting-based combined forecastsa) Number of games. b) In case instruments agree, number of games that predict that outcome. c) Root mean squared error for the deviation between the expected and actual gain of league points for both teams in every game. The combination methods only provide an outcome prediction (home win, draw or away win), from which we derive the expected gain in league points. Thus, the comparability of the RMSE of the combination methods is limited to the RMSE of direct PM and betting odds predictions, which provide separate predictions for the expected gain of league points for each team. d) Profit measured as the (relative) return on betting. e) Total profit (0%) equals the profit realized by betting $100 on each selected gameWe compare the forecast accuracy of different methods, namely, prediction markets, tipsters and betting odds, as well as weighting-based and rule-based combinations of those forecasts. The results indicate that PMs and betting odds yield a comparable and good forecast accuracy. PMs would allow punters to make more money on the betting market if the betting company does not charge fees or at least does not charge monopolistic fees. In contrast, tipsters’ forecasts are poor, in support of the results of previous studies (Forrest & Simmons, 2000). Our findings also support results cited by, among others, Forrest et al. (2005), Boulier et al. (2006) and Spann & Skiera (2003), who show that betting odds and prediction markets provide very good forecasts. 5. Results5.