How good is every NBA player, right now?

Reading DPM

DPM, for Daily Plus Minus, is how much DARKO thinks a player adds to his team's scoring margin, in points per 100 possessions, against an average player. Zero is average. It splits into offense and defense, and on both a higher number is better, so +2 on defense means good defense.

It is a forecast: how DARKO expects a player to play in his next game, from every game he has played, the recent ones counting most. It is not a summary of one season. Here is today's league, one dot per player:

What is +1 worth?

Put a player on a team with four average teammates, at the minutes DARKO projects for him, and the team's rating moves by his DPM times his share of the game. That is how the Roster Lab rates rosters.

How long DARKO remembers

DARKO weighs every game a player has ever played, and each stat forgets at its own speed. It believes a new role within a couple of games and a new shot diet within a few weeks, but a hot shooting month barely moves its view of a shooter: three-point percentage takes about a season and a half to halve. Slide back through the games to see how much one from then still counts.

Inside the model

1. 1 Every game Box scores, tracking data and play-by-play, since 1996-97.
2. 2 A memory for every stat Exponential decay and a Kalman filter weigh every game a player has played.
3. 3 Adjusted For rest, travel and home court, opponents, age, the time of season and team changes.
4. 4 Combined A gradient-boosted decision tree turns them into a projection for every stat.
5. 5 DPM The projections become Box DPM; adding on-off data makes DPM.

DARKO is built using a combination of classical statistical techniques and modern machine learning methods. DARKO is Bayesian in nature, updating its projections in response to new information, with the amount of the update varying by player and by stat, depending on DARKO's confidence in its prior estimate.

The inputs for DARKO are NBA box scores, tracking data, and other game-level information from Basketball-Reference.com, NBA.com, and aided by Darryl Blackport's work in creating pbpstats.com. DARKO's ratings cover every player's games since the 1996-97 season: 788,920 of them so far.

DARKO grapples with the core problem facing every fantasy player (and fan): understanding how much of a given player's development or decline in-season reflects real talent changes, and how much is just the random noise that is part of an NBA season. DARKO addresses these issues without any arbitrary endpoints, i.e., without looking at the last X games of a player's career.

DARKO does this by modeling player performance via an exponential decay model, weighing each game a player has ever played by β^t, where β is some number between 0 and 1, and 't' is the number of days ago a given game took place. The value of β differs for each stat, selected to best predict future results. A differential evolution optimizer is used to calculate each β.

DARKO also combines this exponential decay approach with a modified Kalman filter. Kalman filters are a standard approach used in time-series analysis to model the location of an object for which only noisy measurements are available. Commonly used in fields such as robotics, aviation, rocketry, and neuroscience, it is well suited for sports analysis as well.

A gradient boosted decision tree is used to combine the decay and Kalman projections.

DARKO also accounts for several sports statistics phenomena. These include:

• Rest/Travel/Home Court Effects: As is widely known, players perform worse on the road or on the second night of a back-to-back. DARKO accounts for these effects on a component-by-component level, and the adjustments themselves update daily in response to new information (e.g., home court advantage has been decreasing in the NBA for some time).
• Opponent Adjustments: DARKO's projections account for who each team is playing on a given night, accounting for the projected influence of a player's opponents on each individual stat.
• Aging: DARKO includes an aging curve. Because players improve differently with age in different stats, DARKO uses an independent aging curve for every stat it projects. DARKO also attempts to account for the selection biases which make aging studies very difficult to carry out in sports data.
• Seasonality: Throughout the NBA, offensive efficiency to start the season is usually relatively low league-wide and increases throughout the year. DARKO accounts for a temporary flattening of the rise in assist rates (and other offensive metrics) around the all-star break. All seasonality effects are calculated separately for each component.
• Interaction Effects: DARKO accounts for interactions between various box-score components in making its projections. For example, if a player improves both his three-point shooting and his free throw shooting simultaneously, DARKO will be inherently more credulous of such an improvement.
• Free Agency: Changing teams has a big impact on some box-score components, and DARKO accounts for that. DARKO also gets less confident in its understanding of a player's talent when they change teams, effectively increasing its "learning rate" for these players.

From box scores to DPM

While DARKO is at its core a box-score projection system, it can also be used to generate plus-minus projections, similar in nature to RPM, PIPM, etc. I have called this metric DPM, for "Daily Plus Minus." This metric provides an estimate of how much DARKO thinks each player impacts the score of a game.

Daily Plus Minus is available in two flavors. A box-score-only version (Box DPM), which combines the core box-score metrics to predict player value, and another set (DPM) which adds in on-off data to do the same. Both DPM and Box DPM remain works in progress and may change substantially going forward.

How much the on-off data counts grows with a player's career. Early on DPM is mostly the box score; a few seasons in, mostly on-off, and sooner on defense, where the box score sees less.

The Active Leaderboard also translates DPM and minutes into an annualized Fair Salary / $ Value estimate.

Rookies and the offseason

DARKO has no NCAA, summer league, or preseason data in it. Before a player's first game, his rating comes from his age, draft slot and height, and then DARKO learns about him as he plays, fastest at the start, as Kon Knueppel's season above shows.

Between seasons DARKO keeps projecting every player, pulled back toward average. Until the next season starts, the site shows each player as he stood going into his last game.

Accuracy

While DARKO is not intended to be a DFS tool, given the dearth of other projection systems out there for the NBA, a natural place to test DARKO was to compare how DARKO performs against DFS projections. With one exception, DARKO beat both sites in every stat tested (minutes, points, rebounds, assists, blocks, turnovers, and threes made), some by substantial margins.

The only stat where DARKO lost was in minutes projections. Predictably, playing-time projections are the hardest part of any projection system, and DARKO is no different in this respect.

In HoopsHype's survey of NBA team staff, DPM ranked first among the public all-in-one metrics: 8 of the 29 respondents named it their preferred metric, more than any other; 10 more said they trusted it; 1 said they did not.

“…allowing him to confidently say when a players’ improvement is more signal than noise.”

Cory Jez, the Utah Jazz's former head of analytics, on DPM, to HoopsHype

On the site

• Active Leaderboard Every player's DPM today, with filters, column sets, views you can share and CSV downloads.
• Player pages Career ratings, every season since 1996-97, historical and current-player comps, and a projected box score.
• The Daily What moved in the ratings, game by game, while the season is on.
• Compare Up to four players side by side, head to head.
• Career Trajectories Careers against each other by game, age or season.
• Teams Every team's rating and where it comes from, player by player.
• Standings The standings, with DARKO's season simulations while it is on.
• Roster Lab Trade players, set minutes and see the rating and wins move.
• Fantasy Lab Projected per-game stats under your league's scoring.
• WOWY RAPM Impact from with-or-without data, built the same way back to 1956-57.
• Lineups Two- to five-man units and how they project together.
• Rewind Replay a season week by week, or set the Time Machine to any day since 1996-97.
• Scatterplot Any two numbers against each other.
• Longevity How many seasons each player has left, and when he might retire.
• Rate a Player Two players, one vote: who would you rather have?
• What's new The latest features on the site.

Or ask. Ask DARKO, the search in the menu, takes plain questions like "best defenders under 25", "Jokic vs Wembanyama" or "trade Giannis to the Knicks".

Questions

Glossary

DPM (Daily Plus Minus)

The points per 100 possessions a player adds to his team's margin over an average player, forecast for his next game. Zero is average.

Offense and Defense

DPM's two halves. Higher is better on both.

Box DPM

The version built from box-score projections alone.

On/Off DPM

The version that also uses the team's results with the player on and off the floor.

RAPM

Regularized adjusted plus-minus: team results credited to the players on the floor, shrunk toward average. DARKO's uses play-by-play only, with no box-score prior, and updates daily.

WOWY RAPM

A RAPM-style rating from with-or-without data, built the same way from 1956-57 on.

Projected minutes

DARKO's forecast of a player's minutes a game when he plays.

Fair salary ($ Value)

What a player's on-court value is worth a season, from his DPM and minutes. How it is calculated.

Surplus value

Fair salary minus actual salary. Positive means underpaid.

Origins and thanks

When DARKO started, public basketball stats mostly explained the past. Few projected the future, and fewer did it stat by stat.

DARKO (Daily Adjusted and Regressed Kalman Optimized projections) is an attempt to fill that gap. As will be familiar to baseball fans, DARKO is a basketball projection system similar in concept to Steamer, PECOTA, and ZiPS. To my knowledge, it is one of the few public computer-driven NBA box-score projection systems.

Further, unlike the baseball projection systems listed above (or the CARMELO/RAPTOR projections), DARKO is built from the ground up to update its projections daily, responding to new information as it comes. Instead of just making a projection before the season and leaving users to guess whether a given breakout is "real" or not, DARKO updates its projections for every player in the NBA, for every box-score stat, for every day of the season.

For an audio primer, listen to Kostya on Canzhi Ye's Sports on Paper podcast (January 2022; also on Apple Podcasts): how DARKO works, listener questions about it, and a range of NBA analytics topics, most of them about prediction.

Acknowledgments

Thanks to almost everyone on NBA twitter for help with DARKO's development. Special thanks to Dan Rosenheck, Nathan Walker, and TangoTiger for inspiration in the design of DARKO, and assistance with the underlying math. Thanks to Andrew Patton for building the original Shiny app. Thanks to Ryan Davis for extensive coding assistance, and to Canzhi Ye for scraping assistance. Thanks to Mike Lehrman and Eric Westlund for additional design discussions. Thanks to Krishna Narsu for providing much of the training data used by DARKO. Thanks to Nate Solon for help researching time-series analysis techniques.

Special thanks to Seth Partnow for making sure I didn't just spend all this time building yet-another-all-in-one-stat. And thanks to Crow for making me build one anyway.