Adding the Olympics to an always intense summer schedule increased problems for the tennis warriors in staying healthy and ready for competition. Absences, retirements, and injuries, coupled with a long stretch of rain, hurt the Canadian Open (August 6-13), even as players faced adjustments to hard courts from grass. Some of these problems persisted at the Western and Southern Open in Cincinnati (12-19 August). Still, the Canadian and Cincinnati events offered substantial rewards in ranking points and prize money, even as they added to the evidence pointing toward likely outcomes at U.S. Open.
 
Of the men's Big Four, Rafael Nadal missed both events. Olympics champion Andy Murray withdrew early in Toronto, and Olympics runner-up Roger Federer did not enter. The Canadian crown then went to Novak Djokovic, who lost only one set enroute -- to Tommy Haas, 34, whose strong performance continued his remarkable comeback. Federer then returned to action in Cincinnati, winning strongly over Djokovic in the final there.
 
Meanwhile Petra Kvitova captured the women's crown in Montreal, also against a depleted field, defeating Li Na in a split-set final. Li then won in Cincinnati, beating Kerber, who had beaten both Kvitova and Serena Williams in earlier rounds. The women's 2012 race thus tightened considerably, as no member of the female top four reached the semis at either Canada or Cincinnati.
 
OUTLOOK U.S. OPEN -- MEN'S SINGLES
 
The long dominance of the current Big Four in men's tennis has been unprecedented. One or another member of the Four -- Djokovic, Nadal, Federer, and Murray -- have won 29 of the last 30 Slams. (The only intruder was del Potro, who won U.S. Open in 2009.) So far this year, the Big Four have divided the foremost crowns:
 
Novak Djokovic, Australian Open 2012
Rafael Nadal, Garros 2012
Roger Federer, Wimbledon 2012
Andy Murray, Olympics 2012
 
One year ago at U.S. Open, the Big Four swept past all comers and became the tournament's Final Four. Djokovic and Nadal then met in the final match, where the two staged a ferocious struggle of heavy and aggressively placed rocketry. Djokovic prevailed in four sets, thus capturing his third Slam of 2011.
 
The Final Four at this year's Open will have a different look, as Nadal has withdrawn because of persisting knee trouble. Replacing Rafa in the top-seeded four is Rafa's countryman David Ferrer, who at world rank #5 edges ahead of several others. Following Ferrer are the second four in the seedings -- Tsonga, Berdych, del Potro, and Tipsarevic -- the supposed favorites to join the Big Four in the tournament quarter-finals.
 
Our analysis and prediction begins with two prime indicators.
 
Basic Indicator
 
Our first indicator, Basic Indicator, weights each player's results at tournaments of the past thirteen months. Each tournament's weight depends on how well its past outcomes predicted outcomes at the forthcoming U.S. Open. (The raw values of indicators are hereafter converted to percentages showing each player's probability of winning the Open based only on the stated indicator.)
 
1. Novak Djokovic, 26.8%
2. Roger Federer, 24.1%
3. Andy Murray, 15.9%
4. Jo-wilfried Tsonga, 7.0%
5. Tomas Berdych, 6.6%
6. David Ferrer, 5.3%
7. Juan Martin del Potro, 5.0%
8. Janko Tipsarevic, 2.5%
 
Novak Djokovic's lead in Basic Indicator reflects his triumphs at U.S. Open 2011, Australian Open 2012, and Miami 2012, all very heavily weighted in the calculation. He also won Canada Open 2011 and 2012, moderately weighted here. His recent triumph in Toronto was Novak's first since Miami, however, and that happened without his facing another member of our first eight.
 
Roger Federer meanwhile earned second-best score here while winning Indian Wells, Madrid, and Wimbledon 2012 as well as the year-end championship event in London 2011.
 
Andy Murray's third-best score may undervalue Andy's impressive triumph at the recent Olympics. (In the absence of historical correlation data, a middle-range weight was chosen for the Olympics in our prediction.) Andy then withdrew from Toronto with knee problems and did not play in Cincinnati. Whether the respite from competition will improve Murray's chances at the Open remains unknown.
 
David Ferrer scores below Jo-wilfried Tsonga and Tomas Berdych in this indicator even though he is seeded higher. That is partly because David's successes on clay are less heavily weighted here than in the official rankings and seedings. Tsonga was hurt in an accident at Toronto and then missed Cincinnati but returned as top seed at the final tune-up at Winston-Salem. Ferrer and Berdych both lost early at Cincinnati, beaten by Wawrinka and Raonic, respectively, both of whom played well thereafter, Raonic losing closely to Wawrinka in the fourth round.
 
Federer defeated Djokovic in the final at Cincinnati, both having won their semi-final matches over strong opposition. (Djokovic beat del Potro amid many brutal exchanges, but as the match lengthened del Potro appeared bothered by a left-wrist problem. Meanwhile heavy-hitting Wawrinka kept matters close against Federer.)
 
Quality-Win Quotient, QWQ
 
Our second indicator, Quality-Win Quotient, weights each player's wins over designated elite opponents in 2012 and compares the result against the player's total number of losses to all opponents in the same period. This indicator has sometimes proven highly revealing.
 
1. Roger Federer, 47.8%
2. Novak Djokovic, 32.7%
3. Andy Murray, 14.3%
4. Juan Martin del Potro, 10.6%
5. David Ferrer, 8.5%
6. Tommy Haas, 3.7%
7. Milos Raonic, 3.5%
7. Tomas Berdych, 3.5%
 
Federer, Djokovic, and Murray are almost even in their W-L records against one another. (That would also be true if Nadal were added to the mix.) Our QWQ, however, also includes results against certain lower-ranked players albeit at lowered weights.Thus Federer's six victories against del Potro without loss lift him firmly above Djokovic, who lost to del Potro at the Olympics. Andy Murray's QWQ score is further diminished by his total of losses against all opponents, our divisor here, which is larger than Federer's or Djokovic's.
 
The second tier is also interesting. Del Potro is fourth in QWQ here, ahead of Ferrer, but David actually holds two wins in 2012 over del Potro. However, the tall Argentine's wins over Tsonga, Berdych, Cilic, and Gasquet are more numerous than David's. Tommy Haas's presence here is surprising, he having scored wins this year over Federer, Tsonga, Berdych, and Cilic. Tommy's score is also helped by a comparatively low number of total losses.
 
The Draw and the Predictions
 
The proportion of seeded players who actually attained or exceeded their seeded level at U.S. Open has been slightly over half, according to data from the men's singles starting in 2001. Among the four Slams, seeded players reached their expected round most often at Wimbledon. Garros has been second, and U.S. and Australian Opens are in a virtual tie as the least predictable.
 
The above patterns persisted strongly for reaching the rounds of 32, 16, and 8. But they faltered in recent years when looking at the round of 4, the semi-final round. At the last five U.S. Opens, a total of only four intruders, not seeded in the top four, have actually reached the semis -- i.e., fewer than one per year. In the same period, seven intruders penetrated the final four at Wimbledon, eight each at Garros and Australian Open.
 
Awaiting the official draw, many wondered which of the top two, Federer or Djokovic, would be placed to face Murray in the semis. The answer was Federer, thereby helping Djokovic's apparent chances of reaching the final. Another matter, of much interest among the second tier, was who would be drawn into Ferrer's quarter, presumably lucky in avoiding a Big Four superstar prior to the semis.
 
The predictions that follow are reached by subjective judgment, heavily influenced by Basic Indicator, QWQ, and a third indicator, Head-to-Head Results.
 
Top Quarter
 
It is not exactly a comfortable draw for Roger Federer, whose road to the final four is blocked by two difficult opponents. Mardy Fish has the weapons to extend Roger, and has split sets or forced at least one tiebreaker in all three of their meetings since Mardy's previous head-to-head win, in 2008. They would met in the fourth round of the Open, where Roger's game should have been well honed.
 
The opposite half of the quarter contains Tomas Berdych, the highest seeded, along with Sam Querrey and Nicolas Almagro. Sam is probably the more dangerous for Tomas, the two being almost the same in height, weight, and in playing strengths, Tomas two years the older. Tomas leads Sam in Basic Indicator and QWQ, while Head-to-Head Results provide neutral message. Thus Federer's opponent in the last round of this quarter should be Berdych, whose rank at #5 in Basic Indicator is the highest that Roger might have drawn as his opponent here. Tomas and Roger have split their six meetings of the last three years, where the record shows many split-setters and tiebreak games. But Tomas has tailed downward in his results during the summer, so that if Roger can produce his best, he will prevail.
 
Second Quarter
 
Can Andy Murray produce the same mix of offensive and defensive tennis shown during Wimbledon and the Olympics? He must surmount rising Milos Raonic, while his probable opponent in the final round of this quarter, Jo-wilfried Tsonga, will face the survivor of Cilic vs. Nishikori. Nishikori has the edge in two of our indicators -- QWQ and Head-to-Head Results -- but is slightly behind in Basic Indicator. The edge is Nishikori's. The young Japanese player also shows a five-set win over Jo-wilfried at Australian Open this year. Tsonga's strong lead in Basic Indicator, however, turns our prediction to the powerful and athletic French star.
 
Tsonga missed Cincinnati but competed in Winston-Salem this week, winning his first match there. But Murray leads Tsonga in all three of our indicators, including in head-to-head results, Andy having won their last five meetings although three of these were on grass, which has become Andy's best surface. Amid questions as to Andy's knee and Jo-wilfried's shoulder, Andy should advance to the semis.
 
Third Quarter
 
David Ferrer, highest seeded in this quarter, trails del Potro in QWQ and trails Tsonga and Berdych in Basic Indicator. Any of these might have been drawn into David's quarter. Instead, his most dangerous possible opponents are Tipsarevic, Isner, Gasquet, and Haas. David leads all four in Basic Indicator and QWQ. Head-to-Head Results favor David in all cases, though the record is mixed in all but Gasquet's. Instinct suggests that David is vulnerable, but our indicators show that he must be favored over any opponent in this quarter.
 
Bottom Quarter
 
Djokovic and del Potro are the clear favorites in the opposite halves of this quarter, where del Potro has slightly the more difficult road, with Seppi and Monaco probably ahead. But only if the left-wrist problem seen recently seriously hampers del Potro should the Argentine star fail to advance.
 
But the eventual meeting with Djokovic seems a huge hurdle for the tall Argentine star. Del Potro won the meeting of the two at the Olympics in July, in straight sets. But Djokovic reversed that outcome in Cincinnati two weeks later, where the wrist problem almost surely played a role.
 
Djokovic leads in Basic Indicator and QWQ, though del Potro ranks fourth-best in the latter indicator. Long, brutal rallies are likely to be frequent, where the troublesome wrist will be severely tested. Djokovic should be safe.
 
Semi-finals
 
Novak Djokovic heads David Ferrer in all of our indicators, including in winning both head-to-head meetings in 2012. His victory in their semi-final seems assured.
 
Roger Federer's edge over Andy Murray is far less clear. Andy leads in past head-to-head victories,9-8, including winning three straight sets in their final at the recent Olympics. Roger has won all three of their past meetings at Slams, however, winning nine of the ten sets on these occasions. Roger is of course ahead in our Basic Indicator and QWQ, but Murray is third in both indicators, well ahead of all besides Novak and Roger. Roger will probably be the early attacker, Andy staying in back court using his easier and heavier artillery to forestall Roger's attacking. The choice here is Federer.
 
Final
 
Against Djokovic in their recent final in Cincinnati, Roger played with all guns blazing. For Roger there was little temporizing play, little extending of points and feeling out of his opponent. Having won the first set at love, Roger kept attacking at early opportunity, driving with full power, overspin, and aggressive direction. Novak managed to hold serve throughout the second set. But Roger continued forcing his way forward, keeping Novak deep, consistently out-driving Novak in ground-stroke velocities by 5-10 percent. Novak played well in the concluding tiebreaker, but Roger was finally the winner, Novak having lost a mini-break edge a few points before the finish.
 
Our indicators are contradicting. Novak first in Basic Indicator, Roger first by greater margin in QWQ. The two divided their head-to-head meetings of 2012, Roger winning the last two -- at Wimbledon and Cincinnati -- Novak winning twice on clay earlier.
 
But Roger is now 31, six years older than Novak, who is now at prime age. Physical stamina becomes more critical in best-of-five-set matches. Indeed, Novak has won their last three such meetings on hard courts, including at U.S. Open in 2010 and 2011, where in both cases Novak came from behind to finish ahead. Roger knows how to pace himself, but there are limits to such measures against so strong an opponent. Further, Novak, now forewarned after Cincinnati, should be better prepared to counter Roger's aggression. The choice here to win the tournament is Djokovic.
 
WOMEN'S SINGLES
 
The recent triumphs of Serena Williams at Wimbledon and the Olympics followed a difficult two years in the career of this great champion. Serena hurt her foot in an accident soon after winning Wimbledon 2010, eventually requiring surgery. Then came a serious illness and a dangerous embolism, requiring long recovery. She returned to competition in mid-2011 winning several events and reaching the final at U.S. Open, where she lost, inexplicably dismally.
 
Other, lesser medical problems ensued, seemingly delaying further her full resurgence, which came in London this summer. Indeed, her triumph at the Olympics without loss of a set seemed to place her in a class well above all other current players (when she is able to summon her best). Whatever happens at the Open, Serena, now 30, will be a center of attention there, where she last triumphed in 2008.
 
Also drawing wide attention will be Kim Clijsters, 29, three-time U.S. Open champion. Since becoming runner-up at the Open in 2003, Kim has competed in three U.S. Opens, winning the crown all three times. Intermittent injuries have slowed her career since 2010, but Kim won three matches at the Olympics last month before losing to Sharapova, She has announced that the Open will be her last tournament.
 
Basic Indicator
 
1. Maria Sharapova, 15.9%
2. Agnieszka Radwanska,14.9%
3. Victoria Azarenka, 13.9%
4. Serena Williams, 11.0%
5. Petra Kvitova, 10.9%
6. Caroline Wozniacki, 7.8%
7. Sam Stosur, 7.5%
8. Angelique Kerber, 6.3%
9. Li Na, 5.5%
 
The limited play of Serena Williams since 2010 has produced a seeming undervaluation in this indicator of Serena's chances at the Open. But aside from this reservation, the message seems to be that prospects among the top women are otherwise fairly evenly distributed.
 
Four of the top five -- all save Radwanska -- employ playing styles emphasizing extreme power in serving and stroking. These are the same superstars who have collected the foremost crowns of 2012 to date:
 
Victoria Azarenka, Australian Open 2012
Maria Sharapova, Garros 2012
Serena Williams, Wimbledon and Olympics, 2012
 
It seems clear that the ability to generate, control, and aggressively apply extreme power is key to success at the highest level of women's tennis today. Petra Kvitova, last year's Wimbledon champion, winner at Canada 2012, and fifth in our Basic Indicator here, provides further confirmation. All four -- the three recent Slam winners plus Kvitova -- are able to dominate any other opponent with their serving and stroking power. All have stumbled, however, on occasions when their rockets come with too many errors. Truly great, indeed historic matches can happen when two of them meet if both are able to perform at her best.
 
As to the other leaders in Basic Indicator, above, the playing styles of Li and Stosur are also focused on moderately heavy and selectively aggressive striking. Meanwhile the other three -- Radwanska, Wozniacki, and Kerber -- are more balanced in their tactics, employing extreme power less readily and less likely to provide errors. A fascinating showdown came when Li and Kerber battled in the Cincinnati final recently. Li's late-match run of forceful and error-free striking, the Chinese star having finally managed to curtail her excessive errors, finally overcame Kerber's superb defensive and counter-striking ability. That these two players -- eighth and ninth in Basic Indicator -- reached the final round further suggested that the balance among the sport's top women is becoming tighter.
 
Quality-Win Quotient, QWQ
 
1. Serena Williams, 34.6%
2. Victoria Azarenka, 25.0%
3. Maria Sharapova, 15.5%
4. Venus Williams, 4.4%
5. Li Na, 4.2%
6. Angelique Kerber, 4.1%
7. Kim Clijsters, 3.7%
8. Petra Kvitova, 2.9%
9. Agnieszka Radwanska, 2.6%
 
Serena's top QWQ score here shows her strong dominance in direct meetings with the other leaders. She has won five matches in 2012 against Azarenka and Sharapova, the other players closest-ranked in this indicator, without loss. Meanwhile aside from two walkovers, she has lost only three matches all year against all comers, thus giving her the most favorable divisor of any in this calculation. Even though Serena is only fourth in our first indicator, where her score is hurt by her limited playing activity, it seems clear that the two prime indicators, Basic and QWQ, considered together, make Serena the tournament favorite.
 
Second in QWQ is Victoria Azarenka, lifted by her six match wins this year over Agnieszka Radwanska, without loss. Victoria thus stands well ahead of Maria Sharapova, whom she has beaten in two of their three meetings this year. Maria meanwhile shows three wins over Kvitova and two each over Kerber and Li against a lone loss to these three (to Kerber).
 
The margin after Sharapova is large, and the differences in scores behind Maria are probably not significant. Venus's presence in the second tier here is surprising -- mainly a reflection of her fine performance recently in Cincinnati, where she scored two quality wins and played well in her semi-final loss to Li Na. In that affair Venus displayed her superior power and mobility in much of the going, but Venus's serve was weakened by a back problem that worsened as the match lengthened. Li of course went on to win the tournament. Also in the second tier here is Kim Clijsters. Kim shows four quality wins during the period, which compare well against her total losses of only five.
 
As to Agnieszka, who withdrew from a tune-up tournament with shoulder trouble this week, her year has been spoiled by six losses (and zero wins) against Azarenka -- nearly as many as Agnieszka has lost to all other opponents during the period.
 
The Draw and the Predictions -- Top Quarter
 
Top-seeded in the tournament, Victoria Azarenka has a clear path to the final match of her quarter. Two fine German players -- Goerges and Lisicki -- stand in her way, but neither has consistently performed well enough to be favored over Victoria.
 
The lower half of the quarter is loaded, however, featuring last year's champion Sam Stosur, Cincinnati winner Li Na, and superstar Kim Clijsters. Li and Clijsters should meet in the third round, where Li is favored by Basic Indicator and QWQ. Kim, however, leads in Head-to-Head Results, having won three-setters against Kim in the last two Australian Opens. Na's recent form requires her selection here. But Stosur has beaten Li in all six of their past meetings, including the last three in 2011 on hard courts, and Sam is our choice to reach the final of the quarter.
 
Azarenka is a different matter for Sam, however, Victoria having won all six of their past meetings. With Azarenka ahead in the other indicators as well, her selection here is unproblematic.
 
Second Quarter
 
Two dominating strikers should meet in the final of this quarter. Tall Maria Sharapova holds the head-to-head edge, having won their three meetings in 2012 including their semi-final at Australian Open 2012. Meanwhile Petra Kvitova's lefty serve seems relatively unproblematic for Maria except perhaps on grass. Maria is the more experienced, the more consistent, and her own power should make it hard for Petra to take dominance with her own striking. Maria also has the edge in our first two indicators, so that choosing Maria to prevail here is not difficult.
 
Third Quarter
 
There is no opponent here that Serena Williams cannot dominate with her own power. If she summons full determination and attention to technique -- i.e., her preparatory footwork -- Serena should advance comfortably.
 
Bottom Quarter
 
Venus Williams will bring heavy firepower to her second-round meeting with Angelique Kerber. Angelique leads in Basic Indicator, Venus in QWQ, but the margin is small in both instances. Kerber won both head-to-head meetings in 2012 including at the Olympics in two tiebreak sets. Kerber's rise in the last year has been sensational, while Venus has struggled against illness, producing her best only recently. Kerber, presumably again fresh after an exhausting tournament in Cincinnati, should prevail, answering Venus's power with superb countering and consistency.
 
After that, Kerber will probably meet Errani or perhaps one of the several young risers in her half -- Babos, Bertens, Falconi, McHale, or Muguruza. Sara Errani is the most likely opponent, with the indicators, including a win by Kerber over Sara in their only hard-court meeting this year, favoring Kerber.
 
Finally, representing the lower half of the quarter will be Agnieszka Radwanska. Agnieszka has the edge in Basic Indicator and in Head-to-Head Results, while Angelique leads in QWQ. Agnieszka has shown a tendency consistently to defeat players lower in the rankings than herself, but I still choose Kerber because of her rising performance during the summer compared with Agnieszka's recent pattern of losses.
 
Semi-Finals and Final
 
Since losing to Serena Williams in the semis at Wimbledon, the only appearance by Azarenka has been a withdrawal amid her first set at Montreal prompted by tiredness and a left-knee problem. Sharapova has also been fragile, losing to Lisicki at Wimbledon, demolished by Serena in the final at the Olympics, then withdrawing from Montreal with a viral illness. The indicators are inconclusive, so our choice is a guess. Which superstar is the healthier? The pick here is Sharapova to reach the final.
 
Angelique Kerber defeated Serena in Cincinnati this month, answering Serena power with her own talents in countering. But it is also clear that Serena at her best or close to it should be able to impose her much bigger game on any opponent, including one as clever and able as Kerber. In short, the verdict should rest on Serena's level of play, where it is hard to believe that Serena will not respond to the occasion. The script might resemble Kerber's second-half loss to Li Na in the Cincinnati final, where Na's clean striking at the finish humbled the younger star. Serena should advance.
 
The verdict is also clear for the final match. Sharapova leads Serena in Basic Indicator, while Serena is much farther ahead in QWQ. The crushing evidence is in the head-to-heads. Serena has defeated Maria in 9 of their 11 meetings, including in the last eight. Indeed, in winning their three meetings of 2011-2012, Serena won 36 games, Maria only nine.
 
Serena therefore should achieve this, her fourth U.S. Open crown and her 15th Slam overall.
 
--Ray Bowers, Arlington, Virginia, U.S.A.
 
APPENDIX -- MORE ON THE CALCULATIONS
 
Basic Indicator
 
Each player's results in 25 tournaments of the last 13 months were here weighted. The weights were calculated from historical correlations showing how well the different tournaments have predicted results at the following U.S. Opens. Several other tournaments were not used in the present calculation because their past correlations have been negative. Here are the heaviest weighted events along with their weightings used here for predicting the men's singles at U.S. Open 2012:
 
-- U.S. Open 2011, 8.7%
-- Indian Wells 2012, 7.4%
-- Australian Open 2012, 7.3%
-- Wimbledon 2012, 7.2%
 
The weights used for the women's singles were fairly similar, though Miami 2012 replaces Wimbledon 2012 as one of the four heaviest weighted events.
 
A small correction for player age was used, based on empirical data measuring the greater improvement likely among younger players during the months elapsing between a predictor event and its target. A "random-factor" correction was also made, reflecting that tournaments having fewer entries are for that reason more likely to correlate higher against the target.
 
Quality-Win Quotient, QWQ
 
Match wins in 2012 against the following elite players were weighted triple -- Djokovic, Federer, Nadal, Murray.
 
Wins against the next tier were double-weighted -- Tsonga, Ferrer, Berdych, del Potro.
 
Wins against the third tier were single-weighted -- Tipsarevic, Isner, Almagro, Gasquet, Cilic.
 
Each player's score, obtained above, is then divided by his total losses against all opponents during the period. This gave a raw value of QWQ. The raw values were then adjusted by subtracting the value of the 17th-best score, and these amounts were then converted proportionately so that the total of the top 16 equals 100.
 
The same method is used for the women's singles, where the triple-weighted opponents are Azarenka, Sharapova, A. Radwanska, and S. Williams, the double-weighted ones are Kvitova, Stosur, Wozniacki, and Kerber, and the single-weighted ones are Li, Bartoli, Ivanovic, Errani, and Clijsters.
 
Head-to-Head Results
 
The head-to-head record between each pair of opponents is interpreted subjectively here . (There is no numerical measurement of the record.)