Sunday, April 19, 2015

Spurs' "Hotness" Entering NBA Playoffs

The San Antonio Spurs, owner of five NBA titles including last year's, were floundering for much of the current season, at least relative to their high standards. A four-game losing streak in late February put San Antonio at 34-23. Perhaps having the oldest roster in the NBA was starting to catch up with the Spurs. From that point on, however, Coach Gregg Popovich's crew went 21-4 to finish with a regular-season record of 55-27. And it wasn't just quantity of wins, but also quality, as the Spurs' hot streak included a March 22 win at Atlanta, an April 5 home win over Golden State, and a sweep of an April 8/10 home-and-home match-up vs. Houston.

Baseball-statistics maven Bill James has a statistic he calls "temperature" to assess how hot or cold an individual or team is at the moment. According to this article, the formula adds a standard value to a team's temperature for each win in a streak, regardless of the quality of opposition and other possible features of each win (e.g., home/away, margin of victory). James's temperature for individual baseball players' hotness puts greater weight on recent than distant performance, but it's not clear his team formulas do the same.

I started thinking about a temperature statistic for basketball teams, incorporating quality of opposition (with additional factors such as those listed above possibly being added later). The core concepts are that, against a tough opponent, a win should raise a team's temperature a lot, but a loss shouldn't hurt too much. Conversely, against a weak opponent, a loss should be damaging, but a win not very rewarding.

In my system, a team starts at the neutral point of 1.00. Then, after each game, the previous value is multiplied by an update factor. The multiplier after a win is (1 + opponent's winning percentage), so that the better the opponent, the larger the rise in temperature. The multiplier after a loss is just the opponent's winning percentage, which will drop the temperature (multiplying anything by a number greater than 1.00 increases value, whereas multiplying something by a number between 0.00-1.00 decreases value). The following graphic (on which you can click to enlarge) provides some examples.

The opponent's winning percentage (right before you've played them) appears on the horizontal axis, the red and blue lines are used after a win or loss, respectively, and the multiplier after a game appears on the vertical axis. As one example, suppose your opponent enters the game with a .750 winning percentage and you beat this opponent. The previous value of your "temperature" is then multiplied by 1.750; this is a bigger increase than if you beat a .600 team (which would result in a multiplier of 1.600). Conversely, losing to a .400 teams requires you to multiply your previous temperature by .400, cutting value by more than half (e.g., a previous value of 10 would become 4). Losing to a .800 team, in contrast, doesn't hurt as much (multiplying the previous value by .800).

In order for a win and a loss to cancel each other out and return a team to the neutral point of 1.00, a more dramatic win, such as beating a .750 team, would be offset by losing to a not-quite-as-good team, in this case with a pre-game .571 win percentage, and vice-versa (1.750 x .571 = 1.00, within rounding). The following graph provides a general characterization of the relationship between win and loss multipliers in order to restore a team to 1.00 (neutrality), plus another example.

Enough formulas, let's get to some basketball! First, we see the Spurs' hotness for the final 10 games of each of the past four regular seasons (I think you'll need to click on this chart!).

San Antonio's hotness over its last 10 games of the 2014-15 season is 28.47, obtained by multiplying the automatic start value of 1.00 x 1.685 (for the win over Memphis) x 1.466 (for the win over Miami), and so forth. The season-ending loss to New Orleans (which entered the game with a .543 winning percentage) essentially halved the Spurs' hotness value (i.e., multiplying by .543) in one fell swoop.

The fact that the Spurs' hotness was right around the neutral point of 1.00 in both 2013-14, when they won the NBA championship, and in 2012-13, when only a statistically unlikely comeback by Miami in Game 6 of the finals prevented a San Antonio title, suggests hotness over the final 10 games is not important.Similar findings have been obtained for baseball.

In the lockout-shortened 2011-12 season, however, the Spurs followed up their 10-game winning streak to end the regular season (hotness = 46.75) with 10 straight wins to begin the playoffs, before being eliminated. San Antonio didn't win the title in 2011-12, but a 20-game winning streak spanning the regular season and playoffs is pretty good!

Let's look at some other teams that were hot over their final 10 regular-season games in recent years.

As shown in the top row, the Spurs' opponent in the opening round of this year's playoffs (getting underway tonight), the Los Angeles Clippers, are pretty hot at the moment, too. Both teams are 9-1 over their final 10 regular-season games, but San Antonio (28.47) is hotter than L.A. (18.47), due to the Spurs' higher-quality opposition. For what it's worth, however, the Clippers' 18.47 hotness exceeds the 2012-13 NBA champion Miami Heat's 15.14 in also going 9-1 over its final 10 regular-season games (second row).

Looking at teams with 8-2 records over their final 10 regular-season games this year, the Golden State Warriors, who had far-and-away the NBA's best record (67-15), had a hotness value of 9.76 (third row), and the Boston Celtics, who needed a feverish run just to make the playoffs, had a hotness of 9.65 (last row).

As I noted above, other factors could be added to the mix. Perhaps a team's hotness could be multiplied by bonus adjustment factors of 1.05 or 1.10 (or something else) for each road win or blowout win, or could be multiplied by a deflationary factor of .95 or .90 for a home loss. Recency of performance, which I don't think was a big issue here due to the focus just on teams' final 10 games, could also be taken into account by multiplying newer wins by greater enhancement factors than older wins. Finally, teams' records toward the end of the regular season can be misleading due to resting of players. That's another factor for which adjustments would be helpful. Please share any ideas you have for further refinements, in the Comments section.

Wednesday, April 15, 2015

Korver Faces Tough Odds to Reach 50/50/90 Level

The Atlanta Hawks' Kyle Korver should be familiar to aficionados of hot shooting. The 6-foot-7 shooting guard once had a streak, spanning the 2012-13 and 2013-14 seasons, of making at least one three-pointer in a record 127 straight games (I analyzed Korver's streak here, when it was at 98 games).

During the 2014-15 season, Korver has sought out further frontiers of shooting accuracy. As Ian Levy pointed out back on February 13, Korver was threatening to record the unprecedented feat of hitting 50 percent on all shots from the field, 50 percent from three-point land, and 90 percent on free throws, a so-called 50/50/90 season.

As the Hawks enter their regular-season finale tonight at Chicago, Korver is slightly below all three milestone levels, with a .487 field-goal percentage, .493 three-point percentage, and .897 free-throw percentage (Korver stats page).

It's not even clear how much -- if at all -- Korver will play tonight, as the Hawks rested Korver and other key players last Sunday at Washington, although he played 34 minutes Monday vs. New York. However, assuming he plays tonight, what kind of shooting numbers will he need to post to reach each of the three criteria?

I plotted some equations for how many shots without a miss Korver would need to make to reach .500 on overall field goals and treys, and .900 on free throws. Even if Korver missed a shot of a given type, it would be mathematically possible for him to still reach the milestone, but far more makes and attempts would be necessary than if he never missed.

Let's take three-point shooting, where he enters the game 219 out of 444 (.493). Assuming no missed shots, the number of attempts is equal to the number of makes. We can thus define the equation:

y = (219 + x) / (444 + x)

where y represents Korver's three-point shooting percentage and x represents each new attempt (which is always made). In other words, each new attempt raises his number of attempts beyond the current 444 and each new make raises his number of makes beyond the current 219. By how many attempts (and makes) must x rise to bring y to .500? One can type an equation, such as the one above, into Google, which will automatically generate a plot. Here are the resulting plots for Korver in all three shooting categories (you may click on the graphic to enlarge it).

We see in the upper-right graph that Korver's three-point shooting line (blue upward trend) crosses the .500 threshold (black horizontal line) at six attempts. Six more made threes (again, without a miss) would give him 225, which would be half the new number of attempts, 450. Alternatively, Korver could hit the .500 threshold with a 7-of-8 performance behind the arc, resulting in (226/452). As I said, each miss progressively increases the number of shots he would need to make.  Making 6-of-6 on threes is not terribly likely. Given that his three-point percentage is very close to 50%, let's imagine coin-tossing. Korver would have to flip heads six times in a row, which has a probability of 1-in-64.

Finishing at .900 on free throws should be relatively easy. Korver just needs to make at least three free throws without a miss. If he's not perfect, he would have to make 12 of 13 to reach .900 (117/130).

Lastly, we have overall field-goal percentage. To reach .500, Korver would need a 16-for-16 night (resulting in 305/610) or, alternatively, 17-of-18 (306/612).

Clearly, Korver has his work cut out for him. At the college level, Christian Laettner's performance against Kentucky in the 1992 regional final comes to mind; not only did he hit the turnaround buzzer-beater, but he also hit 10-of-10 from the floor and 10-of-10 from the stripe. Also, Bill Walton hit on 21-of-22 field goals in the 1973 final. That's the kind of game Korver's looking at.

UPDATE: Korver went 3-of-6 from the floor at Chicago to finish the regular season with a .487 field-goal percentage; 2-of-5 on three-pointers for a final percentage of .492 beyond the arc; and 1-of-1 from the free-throw line to finish at .898 from the stripe (box score; final regular-season statistics).