Chapter 11: Regularized Adjusted Plus-Minus (RAPM)

Introduction

The quest to measure individual player value in basketball represents one of the most challenging problems in sports analytics. Unlike baseball, where discrete at-bats provide natural isolation of individual contributions, basketball is a continuous, interconnected game where five players on each team simultaneously influence every possession. A player's raw statistics tell us what they did with the ball, but they cannot capture the countless ways players affect the game without it: setting screens, spacing the floor, rotating on defense, or simply commanding defensive attention.

This chapter introduces Regularized Adjusted Plus-Minus (RAPM), a statistical framework designed to isolate individual player contributions from the complex web of team interactions. RAPM uses regression analysis to estimate each player's impact on team scoring margin while controlling for the quality of teammates and opponents. The "regularized" component addresses fundamental statistical challenges that arise when applying regression to basketball lineup data.

By the end of this chapter, you will understand: - Why naive plus-minus metrics fail to measure true player value - How regression can theoretically isolate individual contributions - Why ordinary least squares produces unreliable estimates for this problem - How ridge regression stabilizes player impact estimates - The complete mathematical framework underlying RAPM - Practical implementation using real NBA data - Extensions including multi-year models and offensive/defensive splits

11.1 The Lineup Problem in Basketball

11.1.1 The Fundamental Challenge

Basketball's fundamental unit of competition is the lineup versus lineup matchup. At any moment, ten players are on the court, and the interaction between these ten players determines what happens. When the Los Angeles Lakers score against the Boston Celtics, that outcome results from:

• The offensive abilities of five Lakers players
• The defensive abilities of five Celtics players
• The synergies (positive or negative) between teammates
• The specific matchups between opposing players
• Randomness inherent in any single possession

Our analytical goal is to decompose observed outcomes into individual player contributions. If we watch a lineup of players A, B, C, D, and E outscore their opponents by 10 points per 100 possessions, how much credit belongs to each player?

11.1.2 The Combinatorial Explosion

Consider the scope of this problem. An NBA team might use 12-15 players throughout a season. The number of possible five-player combinations from 15 players is:

$$\binom{15}{5} = \frac{15!}{5!(15-5)!} = 3,003$$

For a single team. When we consider matchups against opponents, who also have thousands of possible lineups, the combinatorial space explodes. A league with 30 teams, each using 15 players, creates approximately 90,000 possible lineup combinations across all teams. The number of lineup-versus-lineup matchups approaches billions.

Yet a team plays only about 82 games per season, with approximately 200-240 possessions per game. This means roughly 16,000-20,000 possessions of data, during which we observe only a tiny fraction of possible lineup combinations. Most lineup combinations never occur, and those that do often appear for very few possessions.

11.1.3 The Collinearity Problem

Perhaps more problematic than sample size is the structure of the data itself. Players do not appear in random combinations. Starters play with other starters. Bench players play with other bench players. Star players rarely share the court with end-of-bench players.

This creates severe collinearity in our data. When LeBron James always plays with Anthony Davis, their individual effects become nearly impossible to separate statistically. If every possession featuring James also features Davis, how can we determine which player caused the observed scoring margin?

Mathematically, if two players always appear together, their coefficient vectors are perfectly collinear, and no unique solution exists for their individual values. In practice, players exhibit high but imperfect collinearity, leading to unstable estimates that are technically unique but practically unreliable.

11.2 Why Raw Plus-Minus Fails

11.2.1 Definition of Raw Plus-Minus

Raw plus-minus (RPM) is the simplest measure of player impact:

$$\text{Raw PM}_i = \frac{\text{Points Scored} - \text{Points Allowed}}{\text{Possessions}} \times 100$$

for all possessions when player $i$ was on the court. This is typically expressed per 100 possessions to standardize across playing time.

11.2.2 The Teammate Problem

Raw plus-minus conflates individual ability with context. A mediocre player on a championship team will post strong plus-minus numbers simply by sharing the court with excellent teammates. Conversely, an excellent player on a poor team will post weak numbers despite contributing positively.

Consider two hypothetical players: - Player A: Average ability, plays exclusively with four All-Stars - Player B: All-Star ability, plays exclusively with four replacement-level players

Player A's raw plus-minus will likely exceed Player B's, despite Player B being the more valuable individual. The raw metric captures team performance during a player's minutes, not individual contribution.

11.2.3 The Opponent Problem

Raw plus-minus also ignores opponent quality. A player whose minutes come primarily against opposing starters faces tougher competition than a player who enters during garbage time against opposing benches.

If Player C always guards the opponent's best scorer while Player D always guards the weakest offensive player, Player C's defensive plus-minus will suffer despite potentially being the superior defender. The raw metric penalizes players for accepting difficult assignments.

11.2.4 The Sample Size Problem

Raw plus-minus requires large samples to stabilize due to basketball's high variance. A single possession might result in anywhere from 0 to 4+ points, and the outcome depends heavily on random factors (shot-making variance, bounces, referee calls).

The standard error of plus-minus can be approximated using the formula for the standard error of a difference in proportions, modified for basketball's scoring distribution:

$$SE(\text{PM}) \approx \frac{\sigma_{\text{poss}}}{\sqrt{n}} \times 100$$

where $\sigma_{\text{poss}} \approx 2.5$ points per possession and $n$ is the number of possessions. For a player with 1,000 possessions (roughly a full season for a starter), the standard error is approximately:

$$SE \approx \frac{2.5}{\sqrt{1000}} \times 100 \approx 7.9 \text{ points per 100 possessions}$$

This means a player's true value could easily be 8+ points different from their observed raw plus-minus due to random variation alone.

11.3 Introduction to Regression for Player Impact

11.3.1 The Conceptual Framework

Regression offers a path forward by simultaneously modeling all players' contributions. Instead of examining one player at a time, we can estimate all player values jointly while controlling for who else was on the court.

The key insight is treating each possession (or each stint, a continuous period with constant lineups) as an observation. The outcome is the point differential, and the predictors are indicator variables for which players were on the court.

11.3.2 Setting Up the Regression

Let $y_s$ denote the point differential for stint $s$, normalized to per-100-possessions. Define indicator variables:

$$x_{is} = \begin{cases} +1 & \text{if player } i \text{ was on court for team with possession/home team} \\ -1 & \text{if player } i \text{ was on court for opposing/away team} \\ 0 & \text{if player } i \text{ was not on court} \end{cases}$$

The sign convention ensures that positive coefficients always indicate positive contribution to the home/reference team's scoring margin.

Our model becomes:

$$y_s = \beta_0 + \sum_{i=1}^{p} \beta_i x_{is} + \epsilon_s$$

where: - $\beta_0$ is an intercept (often set to 0 or representing home court advantage) - $\beta_i$ is player $i$'s estimated impact per 100 possessions - $\epsilon_s$ is the residual error for stint $s$ - $p$ is the total number of players in the model

11.3.3 Matrix Formulation

In matrix notation, with $n$ stints and $p$ players:

$$\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}$$

where: - $\mathbf{y}$ is an $n \times 1$ vector of stint point differentials - $\mathbf{X}$ is an $n \times p$ design matrix of player indicators - $\boldsymbol{\beta}$ is a $p \times 1$ vector of player coefficients - $\boldsymbol{\epsilon}$ is an $n \times 1$ vector of errors

Each row of $\mathbf{X}$ has exactly 10 non-zero entries (5 players coded as +1 for one team, 5 coded as -1 for the other).

11.3.4 What We Hope to Achieve

If our regression works correctly, each coefficient $\beta_i$ represents player $i$'s marginal contribution to team scoring margin, holding all other players constant. A player with $\beta_i = +3.0$ adds 3 points per 100 possessions compared to a replacement-level baseline (typically set to 0 or the league average).

This automatically controls for teammates and opponents. A player who always plays with stars will have their coefficient estimated relative to those stars' contributions. A player who faces tough opponents will be credited for any positive results against that competition.

11.4 Ordinary Least Squares and Its Problems

11.4.1 The OLS Solution

Ordinary least squares minimizes the sum of squared residuals:

$$\hat{\boldsymbol{\beta}}_{\text{OLS}} = \arg\min_{\boldsymbol{\beta}} \|\mathbf{y} - \mathbf{X}\boldsymbol{\beta}\|^2$$

The solution is given by the normal equations:

$$\hat{\boldsymbol{\beta}}_{\text{OLS}} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}$$

This requires $\mathbf{X}^T\mathbf{X}$ to be invertible.

11.4.2 The Rank Deficiency Problem

In basketball data, $\mathbf{X}^T\mathbf{X}$ is often rank-deficient or nearly so. Several structural issues cause this:

Perfect collinearity: If two players always appear together (or never appear together), their columns in $\mathbf{X}$ are linearly dependent, making $\mathbf{X}^T\mathbf{X}$ singular.

Near-perfect collinearity: Even when players occasionally appear in different combinations, high correlation between player appearance patterns causes $\mathbf{X}^T\mathbf{X}$ to be nearly singular, with very small eigenvalues.

Sum constraint: The ten indicators for on-court players always sum to zero (5 positive, 5 negative), creating a linear dependency among columns.

11.4.3 Ill-Conditioning and Variance Explosion

When $\mathbf{X}^T\mathbf{X}$ is nearly singular, small changes in the data produce large changes in estimated coefficients. The variance of OLS estimates is:

$$\text{Var}(\hat{\boldsymbol{\beta}}_{\text{OLS}}) = \sigma^2(\mathbf{X}^T\mathbf{X})^{-1}$$

When $\mathbf{X}^T\mathbf{X}$ has small eigenvalues, $(\mathbf{X}^T\mathbf{X})^{-1}$ has large eigenvalues, inflating variance enormously.

The condition number of a matrix measures this sensitivity:

$$\kappa(\mathbf{X}^T\mathbf{X}) = \frac{\lambda_{\max}}{\lambda_{\min}}$$

where $\lambda_{\max}$ and $\lambda_{\min}$ are the largest and smallest eigenvalues. Basketball lineup data routinely produces condition numbers exceeding $10^6$ or even $10^{10}$, indicating severe ill-conditioning.

11.4.4 Practical Consequences

When OLS is applied to basketball lineup data:

1. Extreme coefficients: Players may receive estimated values of +50 or -50 points per 100 possessions, far outside any reasonable range.

2. Instability: Adding or removing a few stints dramatically changes all estimates.

3. Meaningless rankings: Player rankings may contradict basic basketball knowledge, with bench warmers ranked above MVP candidates.

4. Wide confidence intervals: Even when point estimates exist, uncertainty is so large as to be useless.

Consider a toy example: if players A and B always appear together and their lineup outscores opponents by +10, OLS might estimate $\beta_A = +100$ and $\beta_B = -90$ (or any combination summing to +10). Both estimates are nonsensical.

11.5 Ridge Regression Fundamentals

11.5.1 The Regularization Concept

Ridge regression addresses ill-conditioning by adding a penalty term that discourages extreme coefficients. Instead of minimizing only the sum of squared residuals, we minimize:

$$\hat{\boldsymbol{\beta}}_{\text{Ridge}} = \arg\min_{\boldsymbol{\beta}} \left[ \|\mathbf{y} - \mathbf{X}\boldsymbol{\beta}\|^2 + \lambda\|\boldsymbol{\beta}\|^2 \right]$$

where $\lambda > 0$ is the regularization parameter (also called the ridge parameter or penalty parameter).

The term $\|\boldsymbol{\beta}\|^2 = \sum_{i=1}^{p} \beta_i^2$ is the squared L2 norm of the coefficient vector. Minimizing this quantity shrinks coefficients toward zero.

11.5.2 The Ridge Solution

Taking the derivative and setting to zero:

$$\frac{\partial}{\partial \boldsymbol{\beta}} \left[ (\mathbf{y} - \mathbf{X}\boldsymbol{\beta})^T(\mathbf{y} - \mathbf{X}\boldsymbol{\beta}) + \lambda\boldsymbol{\beta}^T\boldsymbol{\beta} \right] = 0$$

$$-2\mathbf{X}^T(\mathbf{y} - \mathbf{X}\boldsymbol{\beta}) + 2\lambda\boldsymbol{\beta} = 0$$

$$\mathbf{X}^T\mathbf{X}\boldsymbol{\beta} + \lambda\boldsymbol{\beta} = \mathbf{X}^T\mathbf{y}$$

$$(\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})\boldsymbol{\beta} = \mathbf{X}^T\mathbf{y}$$

Therefore:

$$\hat{\boldsymbol{\beta}}_{\text{Ridge}} = (\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}^T\mathbf{y}$$

The key difference from OLS is adding $\lambda\mathbf{I}$ to $\mathbf{X}^T\mathbf{X}$ before inverting.

11.5.3 Why Ridge Regression Works

Adding $\lambda\mathbf{I}$ to $\mathbf{X}^T\mathbf{X}$ increases all eigenvalues by $\lambda$:

$$\text{eigenvalues of } (\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I}) = \text{eigenvalues of } \mathbf{X}^T\mathbf{X} + \lambda$$

Even if $\mathbf{X}^T\mathbf{X}$ has eigenvalues near zero, $(\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})$ will have eigenvalues at least as large as $\lambda$. This ensures invertibility and reduces the condition number:

$$\kappa(\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I}) = \frac{\lambda_{\max} + \lambda}{\lambda_{\min} + \lambda} < \kappa(\mathbf{X}^T\mathbf{X})$$

As $\lambda$ increases, the condition number approaches 1, completely eliminating ill-conditioning (but at the cost of increased bias).

11.5.4 The Bias-Variance Tradeoff

Ridge regression introduces bias to reduce variance. The expected squared error of an estimator decomposes as:

$$\text{MSE}(\hat{\boldsymbol{\beta}}) = \text{Bias}^2(\hat{\boldsymbol{\beta}}) + \text{Var}(\hat{\boldsymbol{\beta}})$$

OLS is unbiased but may have enormous variance due to ill-conditioning. Ridge regression is biased (it shrinks coefficients toward zero) but dramatically reduces variance.

For small $\lambda$, the variance reduction typically exceeds the bias increase, reducing total MSE. As $\lambda$ grows very large, bias dominates and MSE increases again. There exists an optimal $\lambda$ that minimizes MSE.

The bias of ridge regression is:

$$\text{Bias}(\hat{\boldsymbol{\beta}}_{\text{Ridge}}) = -\lambda(\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})^{-1}\boldsymbol{\beta}$$

This shows that ridge bias shrinks true coefficients toward zero. Players who truly have large (positive or negative) impacts will have their estimates attenuated.

11.5.5 Bayesian Interpretation

Ridge regression has an elegant Bayesian interpretation. If we assume:

1. A linear model: $\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}$ where $\boldsymbol{\epsilon} \sim N(0, \sigma^2\mathbf{I})$
2. A prior on coefficients: $\boldsymbol{\beta} \sim N(0, \tau^2\mathbf{I})$

Then the posterior mode (MAP estimate) is:

$$\hat{\boldsymbol{\beta}}_{\text{MAP}} = (\mathbf{X}^T\mathbf{X} + \frac{\sigma^2}{\tau^2}\mathbf{I})^{-1}\mathbf{X}^T\mathbf{y}$$

This is exactly ridge regression with $\lambda = \frac{\sigma^2}{\tau^2}$.

The Bayesian interpretation is intuitive for basketball: we have a prior belief that most players have values near zero (near league average), and we update this belief based on observed data. When data is limited or collinear, we rely more heavily on the prior. The ratio $\frac{\sigma^2}{\tau^2}$ controls how strongly we shrink toward the prior versus trusting the data.

11.6 Building a RAPM Model Step-by-Step

11.6.1 Data Preparation

The first step is preparing stint-level data. A stint is a continuous period of play with the same ten players on the court. When any substitution occurs, a new stint begins.

For each stint $s$, we need: - Point differential: $y_s = \text{Home Points} - \text{Away Points}$ - Possessions: $\text{poss}_s$ (for weighting and normalization) - Player indicators: $x_{is}$ for each player $i$

Points differential should be normalized per 100 possessions:

$$y_s^{\text{norm}} = y_s \times \frac{100}{\text{poss}_s}$$

11.6.2 Design Matrix Construction

Create the design matrix $\mathbf{X}$ where: - Each row represents one stint - Each column represents one player - Entry $(s, i) = +1$ if player $i$ played for home team in stint $s$ - Entry $(s, i) = -1$ if player $i$ played for away team in stint $s$ - Entry $(s, i) = 0$ otherwise

Note: Some implementations use $\{0, 1\}$ coding with separate columns for each team's players. The $\{-1, 0, +1\}$ coding is more parsimonious and directly interpretable.

11.6.3 Weighting by Possessions

Stints vary dramatically in length, from single possessions to 10+ minutes. Longer stints provide more information and should receive more weight.

The standard approach is weighted least squares with weights proportional to possessions:

$$w_s = \text{poss}_s$$

This is equivalent to using possession-level observations but is computationally more efficient.

In matrix form, define $\mathbf{W} = \text{diag}(w_1, \ldots, w_n)$. The weighted ridge regression problem becomes:

$$\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{W}\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}^T\mathbf{W}\mathbf{y}$$

11.6.4 Choosing the Regularization Parameter

Selecting $\lambda$ is critical. Too small, and coefficients remain unstable. Too large, and all players shrink toward zero, eliminating meaningful differences.

Common approaches:

Cross-validation: Divide data into folds, train on subsets, evaluate prediction error on held-out data. Choose $\lambda$ minimizing average test error.

$$\lambda^* = \arg\min_\lambda \sum_{k=1}^{K} \|\mathbf{y}^{(k)} - \mathbf{X}^{(k)}\hat{\boldsymbol{\beta}}_{-k}(\lambda)\|^2$$

where $\hat{\boldsymbol{\beta}}_{-k}(\lambda)$ is estimated excluding fold $k$.

Generalized cross-validation (GCV): An efficient leave-one-out approximation:

$$\text{GCV}(\lambda) = \frac{1}{n} \frac{\|\mathbf{y} - \mathbf{X}\hat{\boldsymbol{\beta}}(\lambda)\|^2}{(1 - \text{tr}(\mathbf{H}_\lambda)/n)^2}$$

where $\mathbf{H}_\lambda = \mathbf{X}(\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}^T$ is the hat matrix.

Prior knowledge: In practice, analysts often use $\lambda$ values calibrated from prior research. Typical values for single-season NBA RAPM fall in the range $\lambda \in [1000, 10000]$, depending on data scaling.

11.6.5 Fitting the Model

With data prepared and $\lambda$ selected, fitting is straightforward:

import numpy as np
from scipy import linalg

def fit_rapm(X, y, weights, lambda_reg):
    """
    Fit RAPM model using weighted ridge regression.

    Parameters:
    -----------
    X : array (n_stints, n_players)
        Design matrix with player indicators
    y : array (n_stints,)
        Point differential per 100 possessions
    weights : array (n_stints,)
        Possession weights for each stint
    lambda_reg : float
        Ridge regularization parameter

    Returns:
    --------
    beta : array (n_players,)
        Estimated player coefficients (RAPM values)
    """
    # Create weight matrix
    W = np.diag(weights)

    # Compute weighted cross-product
    XtWX = X.T @ W @ X
    XtWy = X.T @ W @ y

    # Add ridge penalty
    n_players = X.shape[1]
    ridge_matrix = XtWX + lambda_reg * np.eye(n_players)

    # Solve system
    beta = linalg.solve(ridge_matrix, XtWy)

    return beta

11.6.6 Using scikit-learn

For production use, scikit-learn provides optimized implementations:

from sklearn.linear_model import Ridge

def fit_rapm_sklearn(X, y, weights, lambda_reg):
    """
    Fit RAPM using scikit-learn's Ridge regression.

    Note: scikit-learn uses alpha = lambda / sum(weights) for weighted data
    """
    # Adjust lambda for sklearn's scaling
    alpha = lambda_reg

    # Fit ridge model
    model = Ridge(alpha=alpha, fit_intercept=False)
    model.fit(X, y, sample_weight=weights)

    return model.coef_

11.7 Interpreting RAPM Coefficients

11.7.1 The Basic Interpretation

Each RAPM coefficient $\beta_i$ represents player $i$'s estimated contribution to team scoring margin per 100 possessions, relative to replacement level (typically set to league average, which is zero).

If $\beta_i = +4.5$, player $i$ is estimated to improve their team's scoring margin by 4.5 points per 100 possessions compared to a replacement-level player. Over a full game (approximately 100 possessions), this player contributes roughly 4.5 extra points to victory margin.

11.7.2 The Replacement Level Question

RAPM coefficients are relative to a baseline. The most common baseline is league average, where the average coefficient is zero. This means: - $\beta_i > 0$: Player contributes above average - $\beta_i < 0$: Player contributes below average - $\beta_i = 0$: Player contributes at average level

Some analysts prefer replacement level (the value of a freely available player) as the baseline. This requires adjusting all coefficients:

$$\beta_i^{\text{replacement}} = \beta_i - \beta_{\text{replacement}}$$

where $\beta_{\text{replacement}}$ is typically estimated around -2 to -3 per 100 possessions.

11.7.3 Converting to Wins

RAPM values can be converted to estimated wins using the relationship between point differential and win percentage:

$$\text{Expected Wins} = 41 + \frac{\text{Season Point Differential}}{2.7}$$

For a player with RAPM = $\beta$ playing $M$ minutes in a season:

$$\text{RAPM Wins} = \beta \times \frac{M}{\text{minutes per 100 possessions}} \times \frac{1}{2.7}$$

With approximately 2.0 minutes per 100 possessions (varies by pace):

$$\text{RAPM Wins} \approx \beta \times \frac{M}{200 \times 2.7} \approx \beta \times \frac{M}{540}$$

An elite player with $\beta = 5.0$ playing 2,500 minutes contributes approximately:

$$5.0 \times \frac{2500}{540} \approx 23 \text{ wins above replacement}$$

11.7.4 Confidence and Uncertainty

RAPM point estimates come with substantial uncertainty, particularly for players with limited playing time or unusual lineup patterns. While exact standard errors require bootstrap or Bayesian computation, rough guidelines include:

• High certainty: Starters with 2,000+ minutes, consistent lineup patterns
• Moderate certainty: Rotation players with 1,000-2,000 minutes
• Low certainty: Bench players with < 1,000 minutes, unusual lineup patterns

The standard error of ridge estimates is:

$$\text{Var}(\hat{\boldsymbol{\beta}}_{\text{Ridge}}) = \sigma^2 (\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})^{-1} \mathbf{X}^T\mathbf{X} (\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})^{-1}$$

This is complex to compute but can be approximated via bootstrap resampling.

11.7.5 What RAPM Captures (and Doesn't)

RAPM captures all aspects of player impact that affect scoring margin while on court: - Scoring and shot creation - Defensive stops - Rebounding (offensive and defensive) - Playmaking - Spacing effects - Screen setting - Off-ball movement - Defensive communication and help - "Gravity" effects on teammates

RAPM does not capture: - Impact when off the court (coaching, leadership, practice) - Rest effects on teammates - Long-term development effects - Playoff-specific performance (unless using playoff data) - Injury risk or durability

11.8 Multi-Year RAPM and Using Priors

11.8.1 The Value of Multiple Seasons

Single-season RAPM suffers from sample size limitations. Even with regularization, estimates remain noisy. Using multiple seasons of data dramatically improves reliability by: - Increasing total observations - Observing more lineup combinations - Reducing impact of single-season anomalies

Multi-year RAPM (often called multi-year regularized adjusted plus-minus, MY-RAPM) pools data across 2-5 seasons.

11.8.2 Implementation Approaches

Simple pooling: Combine all stints from multiple seasons into a single dataset and fit standard RAPM. This implicitly assumes player ability is constant across seasons.

Year-weighted pooling: Weight recent seasons more heavily than distant seasons to account for player development and aging:

$$w_s^{\text{total}} = w_s^{\text{poss}} \times \gamma^{(\text{current\_year} - \text{stint\_year})}$$

where $\gamma \in (0, 1)$ is a decay factor (e.g., $\gamma = 0.8$ reduces prior year weight by 20%).

Year-specific coefficients with regularization: Estimate separate coefficients for each player-year but regularize toward each other:

$$\min_{\boldsymbol{\beta}} \sum_t \|\mathbf{y}_t - \mathbf{X}_t\boldsymbol{\beta}_t\|^2 + \lambda_1 \sum_t \|\boldsymbol{\beta}_t\|^2 + \lambda_2 \sum_t \|\boldsymbol{\beta}_t - \boldsymbol{\beta}_{t-1}\|^2$$

11.8.3 Informative Priors

The Bayesian interpretation of ridge regression suggests using informative priors rather than simply shrinking toward zero. Instead of assuming all players have prior mean zero, we can incorporate:

Position-based priors: Different prior means for different positions based on historical patterns.

Box score-based priors: Use box score statistics (points, rebounds, assists, etc.) to form prior estimates. This is the approach used in RPM (Real Plus-Minus) and similar hybrid metrics.

The prior-augmented ridge regression becomes:

$$\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{W}\mathbf{X} + \lambda\mathbf{I})^{-1}(\mathbf{X}^T\mathbf{W}\mathbf{y} + \lambda\boldsymbol{\mu})$$

where $\boldsymbol{\mu}$ is the vector of prior means.

11.8.4 Box Score Prior (BPM) Integration

ESPN's RPM and similar metrics use box score statistics to form informative priors. The process:

1. Build a model predicting RAPM from box score stats using historical data
2. For each player, predict their expected RAPM from box scores: $\mu_i = f(\text{box score}_i)$
3. Use these predictions as prior means in the regularized regression

This hybrid approach combines: - Box scores: Stable, available for all players, capture counting stats - Plus-minus: Captures non-box-score contributions, adjusts for context

The optimal weight between prior and data depends on sample size and prior quality. With little playing time, the estimate relies heavily on the box score prior. With extensive playing time, actual lineup data dominates.

11.9 Offensive and Defensive RAPM Splits

11.9.1 Motivation for Splits

Overall RAPM collapses offense and defense into a single number. But a player with RAPM = +2 could be: - An elite offensive player (+5) and poor defender (-3) - A poor offensive player (-3) and elite defender (+5) - Average at both offense (+1) and defense (+1)

Understanding the split helps with lineup construction, game planning, and player evaluation.

11.9.2 Separate Models for O-RAPM and D-RAPM

The simplest approach estimates separate models:

Offensive RAPM (O-RAPM): Model predicting team offensive efficiency (points scored per 100 possessions) from player indicators. Each coefficient represents contribution to team offense.

Defensive RAPM (D-RAPM): Model predicting team defensive efficiency (points allowed per 100 possessions) from player indicators. Note the sign convention: positive D-RAPM means the player makes the defense worse (allows more points).

For interpretability, D-RAPM is often negated so that positive values indicate good defense:

$$\text{D-RAPM}_{\text{reported}} = -\text{D-RAPM}_{\text{raw}}$$

Then: $\text{Total RAPM} = \text{O-RAPM} + \text{D-RAPM}_{\text{reported}}$

11.9.3 Implementation Details

For each stint, we now have two outcomes: - $y_s^O$: Points scored (by home team) per 100 possessions - $y_s^D$: Points allowed (by home team) per 100 possessions

The models are:

$$y_s^O = \sum_i \beta_i^O x_{is} + \epsilon_s^O$$

$$y_s^D = \sum_i \beta_i^D x_{is} + \epsilon_s^D$$

These can be fit separately using the same ridge regression framework. The design matrix $\mathbf{X}$ is identical; only the response vector differs.

def fit_orapm_drapm(X, y_off, y_def, weights, lambda_reg):
    """
    Fit separate offensive and defensive RAPM models.

    Parameters:
    -----------
    y_off : array (n_stints,)
        Offensive rating (points scored per 100 poss)
    y_def : array (n_stints,)
        Defensive rating (points allowed per 100 poss)

    Returns:
    --------
    orapm : array (n_players,)
        Offensive RAPM values
    drapm : array (n_players,)
        Defensive RAPM values (positive = good defense)
    """
    from sklearn.linear_model import Ridge

    model = Ridge(alpha=lambda_reg, fit_intercept=False)

    # Offensive model
    model.fit(X, y_off, sample_weight=weights)
    orapm = model.coef_

    # Defensive model
    model.fit(X, y_def, sample_weight=weights)
    drapm_raw = model.coef_

    # Negate so positive = good defense
    drapm = -drapm_raw

    return orapm, drapm

11.9.4 Challenges with Splits

Splitting introduces additional noise. The variance of O-RAPM and D-RAPM individually exceeds the variance of total RAPM because:

1. Each model uses half the variation in outcomes
2. Offensive and defensive success are correlated (good teams do both)
3. Sample size constraints are unchanged

Additionally, the "correct" split is philosophically ambiguous. Is a center's offensive rebound offensive value (second-chance points) or defensive value (preventing opponent fast breaks)? Does a guard's help defense that allows an open three constitute offensive value (they were focused on offense) or defensive value (they failed defensively)?

11.9.5 Interpreting Splits

Despite challenges, O-RAPM and D-RAPM provide useful information:

Lineup construction: Pair offensive and defensive specialists to create balanced units.

Matchup planning: Deploy defensive stoppers against opposing scorers; hide weak defenders against poor offensive opponents.

Contract evaluation: Offensive skills may age differently than defensive skills, informing long-term value projections.

Draft evaluation: Assess whether prospects' box score production reflects genuine offensive skill or merely opportunity.

11.10 Comparison to Other Impact Metrics

11.10.1 Overview of Plus-Minus Family

The landscape of plus-minus metrics includes:

11.10.2 Box Score-Based Alternatives

Box Plus-Minus (BPM): Estimates plus-minus from box score statistics using regression coefficients derived from historical RAPM. Advantages: Stable, interpretable, available for any player with box scores. Disadvantages: Misses non-box-score contributions, biased toward certain play styles.

Win Shares: Allocates team wins to players based on offensive and defensive contributions estimated from box scores. Advantages: Intuitive "wins" scale, comprehensive. Disadvantages: Assumes team success is fully allocable to individuals.

PER (Player Efficiency Rating): Weighted sum of box score statistics, normalized to league average (15). Advantages: Simple, widely available. Disadvantages: Heavily favors volume scorers, ignores defense, no lineup adjustment.

11.10.3 Tracking-Based Metrics

Modern NBA tracking data enables new metrics:

Defensive RAPTOR: Incorporates tracking-derived defensive metrics (contests, deflections, etc.) as priors.

DRAYMOND (Defensive Rating Accounting for Yielding Minimal Openness by Naismith Defense): Uses tracking data to measure defensive positioning and shot-altering ability.

Luck-Adjusted Ratings: Adjust for three-point variance using tracking-derived shot quality data.

11.10.4 RAPM vs. Alternatives

Advantages of RAPM: - Theoretically measures true impact, not proxies - Captures everything affecting scoring margin - Adjusts for teammates and opponents - No assumptions about which actions matter

Disadvantages of RAPM: - Requires large samples for reliability - Biased toward zero due to regularization - Cannot explain why a player is valuable - Sensitive to regularization parameter choice - Does not capture off-court value

11.10.5 Hybrid Approaches

State-of-the-art metrics combine approaches:

1. Use box scores and tracking data to form informative priors
2. Apply regularized regression to lineup data
3. Weight prior versus data based on sample size
4. Report uncertainty intervals alongside point estimates

ESPN's RPM, FiveThirtyEight's RAPTOR, and similar metrics follow this template. The key insight is that no single data source is sufficient; combining complementary information yields better estimates.

11.11 Complete Mathematical Derivation

11.11.1 The Full Model Specification

Let us formalize the complete RAPM model:

Observations: $n$ stints indexed by $s = 1, \ldots, n$

Players: $p$ players indexed by $i = 1, \ldots, p$

Design Matrix: $\mathbf{X} \in \mathbb{R}^{n \times p}$ where

$$X_{si} = \begin{cases} +1 & \text{if player } i \text{ on court for home team in stint } s \\ -1 & \text{if player } i \text{ on court for away team in stint } s \\ 0 & \text{otherwise} \end{cases}$$

Response: $\mathbf{y} \in \mathbb{R}^n$ where $y_s$ is the point differential (home - away) per 100 possessions for stint $s$

Weights: $\mathbf{w} \in \mathbb{R}^n_+$ where $w_s$ is the number of possessions in stint $s$

Model: $\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}$ where $\epsilon_s \overset{ind}{\sim} N(0, \sigma^2/w_s)$

Prior: $\boldsymbol{\beta} \sim N(\boldsymbol{\mu}, \tau^2 \mathbf{I})$

11.11.2 Derivation of the Posterior Mode

The log-posterior (ignoring constants) is:

$$\log p(\boldsymbol{\beta} | \mathbf{y}) \propto -\frac{1}{2\sigma^2} \sum_s w_s (y_s - \mathbf{x}_s^T\boldsymbol{\beta})^2 - \frac{1}{2\tau^2} \|\boldsymbol{\beta} - \boldsymbol{\mu}\|^2$$

In matrix notation:

$$\log p(\boldsymbol{\beta} | \mathbf{y}) \propto -\frac{1}{2\sigma^2} (\mathbf{y} - \mathbf{X}\boldsymbol{\beta})^T \mathbf{W} (\mathbf{y} - \mathbf{X}\boldsymbol{\beta}) - \frac{1}{2\tau^2} (\boldsymbol{\beta} - \boldsymbol{\mu})^T(\boldsymbol{\beta} - \boldsymbol{\mu})$$

where $\mathbf{W} = \text{diag}(w_1, \ldots, w_n)$.

Taking the gradient with respect to $\boldsymbol{\beta}$:

$$\nabla_{\boldsymbol{\beta}} \log p = \frac{1}{\sigma^2} \mathbf{X}^T\mathbf{W}(\mathbf{y} - \mathbf{X}\boldsymbol{\beta}) - \frac{1}{\tau^2}(\boldsymbol{\beta} - \boldsymbol{\mu})$$

Setting to zero:

$$\frac{1}{\sigma^2} \mathbf{X}^T\mathbf{W}\mathbf{y} - \frac{1}{\sigma^2} \mathbf{X}^T\mathbf{W}\mathbf{X}\boldsymbol{\beta} - \frac{1}{\tau^2}\boldsymbol{\beta} + \frac{1}{\tau^2}\boldsymbol{\mu} = 0$$

$$\left(\frac{1}{\sigma^2} \mathbf{X}^T\mathbf{W}\mathbf{X} + \frac{1}{\tau^2}\mathbf{I}\right)\boldsymbol{\beta} = \frac{1}{\sigma^2} \mathbf{X}^T\mathbf{W}\mathbf{y} + \frac{1}{\tau^2}\boldsymbol{\mu}$$

Multiplying both sides by $\sigma^2$:

$$\left(\mathbf{X}^T\mathbf{W}\mathbf{X} + \frac{\sigma^2}{\tau^2}\mathbf{I}\right)\boldsymbol{\beta} = \mathbf{X}^T\mathbf{W}\mathbf{y} + \frac{\sigma^2}{\tau^2}\boldsymbol{\mu}$$

Defining $\lambda = \frac{\sigma^2}{\tau^2}$:

$$\hat{\boldsymbol{\beta}}_{\text{MAP}} = \left(\mathbf{X}^T\mathbf{W}\mathbf{X} + \lambda\mathbf{I}\right)^{-1}\left(\mathbf{X}^T\mathbf{W}\mathbf{y} + \lambda\boldsymbol{\mu}\right)$$

When $\boldsymbol{\mu} = \mathbf{0}$ (standard RAPM):

$$\hat{\boldsymbol{\beta}}_{\text{RAPM}} = \left(\mathbf{X}^T\mathbf{W}\mathbf{X} + \lambda\mathbf{I}\right)^{-1}\mathbf{X}^T\mathbf{W}\mathbf{y}$$

11.11.3 Posterior Variance

The posterior distribution is Gaussian:

$$\boldsymbol{\beta} | \mathbf{y} \sim N(\hat{\boldsymbol{\beta}}_{\text{MAP}}, \boldsymbol{\Sigma}_{\text{post}})$$

where:

$$\boldsymbol{\Sigma}_{\text{post}} = \sigma^2 \left(\mathbf{X}^T\mathbf{W}\mathbf{X} + \lambda\mathbf{I}\right)^{-1}$$

This provides uncertainty quantification. The posterior standard deviation for player $i$ is:

$$\text{SD}(\beta_i | \mathbf{y}) = \sqrt{\sigma^2 \left[(\mathbf{X}^T\mathbf{W}\mathbf{X} + \lambda\mathbf{I})^{-1}\right]_{ii}}$$

11.11.4 Cross-Validation Derivation

Leave-one-out cross-validation (LOOCV) error can be computed efficiently using the Sherman-Morrison formula. The LOOCV estimate for stint $s$ is:

$$\hat{y}_{-s} = \mathbf{x}_s^T \hat{\boldsymbol{\beta}}_{-s}$$

where $\hat{\boldsymbol{\beta}}_{-s}$ is estimated excluding stint $s$. The LOOCV error is:

$$\text{LOOCV}(\lambda) = \sum_s w_s (y_s - \hat{y}_{-s})^2$$

Using matrix algebra:

$$y_s - \hat{y}_{-s} = \frac{y_s - \hat{y}_s}{1 - h_{ss}}$$

where $\hat{y}_s = \mathbf{x}_s^T \hat{\boldsymbol{\beta}}$ and $h_{ss}$ is the $s$-th diagonal element of the hat matrix:

$$\mathbf{H}_\lambda = \mathbf{X}(\mathbf{X}^T\mathbf{W}\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}^T\mathbf{W}$$

This allows computing LOOCV without refitting the model $n$ times.

11.11.5 Eigenvalue Decomposition Perspective

Let $\mathbf{X} = \mathbf{U}\mathbf{D}\mathbf{V}^T$ be the singular value decomposition of $\mathbf{X}$, where: - $\mathbf{U}$ is $n \times r$ with orthonormal columns (stint loadings) - $\mathbf{D}$ is $r \times r$ diagonal with singular values $d_1 \geq d_2 \geq \ldots \geq d_r > 0$ - $\mathbf{V}$ is $p \times r$ with orthonormal columns (player loadings) - $r = \text{rank}(\mathbf{X})$

The ridge solution can be written:

$$\hat{\boldsymbol{\beta}}_{\text{Ridge}} = \mathbf{V} \text{diag}\left(\frac{d_j}{d_j^2 + \lambda}\right) \mathbf{U}^T \mathbf{y}$$

This shows that ridge regression: 1. Projects data onto principal components of $\mathbf{X}$ 2. Shrinks each component by a factor $\frac{d_j}{d_j^2 + \lambda}$ 3. Components with small singular values (high collinearity) are shrunk most heavily

11.12 Advanced Topics

11.12.1 Heterogeneous Regularization

Standard RAPM applies the same regularization to all players. But players with more playing time have better-estimated coefficients and need less shrinkage. Heterogeneous regularization uses different $\lambda_i$ for each player:

$$\min_{\boldsymbol{\beta}} \|\mathbf{y} - \mathbf{X}\boldsymbol{\beta}\|^2_{\mathbf{W}} + \sum_i \lambda_i \beta_i^2$$

Players with extensive minutes receive small $\lambda_i$; players with limited minutes receive large $\lambda_i$.

Implementation uses a diagonal penalty matrix:

$$\hat{\boldsymbol{\beta}} = (\mathbf{X}^T\mathbf{W}\mathbf{X} + \boldsymbol{\Lambda})^{-1}\mathbf{X}^T\mathbf{W}\mathbf{y}$$

where $\boldsymbol{\Lambda} = \text{diag}(\lambda_1, \ldots, \lambda_p)$.

11.12.2 Time-Varying Coefficients

Player ability changes within and across seasons. Time-varying RAPM models allow coefficients to evolve:

$$\beta_i(t) = \beta_i(t-1) + \eta_i(t)$$

where $\eta_i(t) \sim N(0, \sigma_\eta^2)$ is random innovation.

This state-space formulation can be estimated using Kalman filtering or Bayesian smoothing. It produces a trajectory of player value over time rather than a single number.

11.12.3 Interaction Effects

Standard RAPM assumes additive player effects: lineup value equals the sum of individual values. But basketball involves synergies and redundancies. Some player pairs complement each other; others clash.

Interaction RAPM adds pairwise terms:

$$y_s = \sum_i \beta_i x_{is} + \sum_{i < j} \gamma_{ij} x_{is} x_{js} + \epsilon_s$$

The challenge is the explosion of parameters. With 500 players, there are $\binom{500}{2} = 124,750$ possible interactions. Strong regularization (potentially including Lasso for sparsity) is essential.

11.12.4 Lineup-Level Features

Beyond player indicators, models can incorporate lineup-level features: - Height distribution - Position composition (number of guards, wings, bigs) - Pace tendencies - Experience levels - Rest days

These features help explain variance and improve player coefficient stability.

11.12.5 Possession-Level Modeling

Rather than using stint-level point differentials, some models work at the possession level:

$$P(\text{score} | \text{lineup}_{\text{off}}, \text{lineup}_{\text{def}}) = f\left(\sum_i \beta_i^O x_i^{\text{off}} - \sum_j \beta_j^D x_j^{\text{def}}\right)$$

Logistic or multinomial regression models the probability of scoring (or the distribution of points scored). This provides a cleaner link function and properly handles the discrete nature of scoring.

11.13 Practical Considerations

11.13.1 Data Sources

Implementing RAPM requires play-by-play data with: - Timestamp or possession identifier for each event - Complete lineup information (all 10 players) - Scoring events with point values - Game and season identifiers

Public sources include: - Basketball-Reference (historical play-by-play) - NBA Stats API (current season data) - Kaggle datasets (compiled historical data)

Commercial sources provide cleaner, more detailed data but require licensing agreements.

11.13.2 Data Quality Issues

Common problems include: - Missing lineup data: Substitutions not recorded, requiring inference - Inconsistent player identification: Name variations, traded players - Edge cases: Technical fouls, delays, replays affecting timing - Era differences: Three-point line changes, pace variation

Careful data cleaning is essential. Validate totals against official box scores.

11.13.3 Computational Considerations

For modern NBA seasons: - ~250,000 stints per season - ~500 players per season - Design matrix is 250,000 x 500

Ridge regression requires solving a 500 x 500 system, which is trivial for modern computers. The main computational cost is data preparation and cross-validation.

For multi-year models or interaction effects, computation becomes more substantial. Sparse matrix representations and iterative solvers may be necessary.

11.13.4 Reporting and Communication

When reporting RAPM: 1. Specify the time period (single season, multi-year, career) 2. Indicate the regularization approach (ridge, Lasso, elastic net) 3. Report uncertainty if possible (confidence intervals, posterior distributions) 4. Provide context (league average, percentile rank) 5. Include sample size (minutes, possessions)

Avoid overinterpreting small differences. Two players with RAPM values of +2.1 and +1.9 are statistically indistinguishable.

11.14 Summary

Regularized Adjusted Plus-Minus represents a principled approach to measuring player value in basketball. By framing the problem as regression, we can theoretically isolate individual contributions from team contexts. Ridge regression addresses the collinearity that makes ordinary least squares fail.

Key takeaways:

1. Raw plus-minus conflates player ability with context. Without adjustment, we cannot separate individual contribution from teammate/opponent effects.

2. Regression-based approaches control for context by simultaneously estimating all players' effects conditional on who else was on the court.

3. Collinearity makes OLS unstable. Players appear in correlated patterns, creating near-singular design matrices and explosive variance.

4. Ridge regression stabilizes estimates by penalizing extreme coefficients. This introduces bias but dramatically reduces variance.

5. The regularization parameter balances bias and variance. Cross-validation or prior knowledge guides selection.

6. RAPM coefficients represent impact per 100 possessions relative to a baseline (typically league average).

7. Multi-year models and informative priors improve reliability by increasing effective sample size and incorporating additional information.

8. Offensive and defensive splits provide insight but are noisier than overall estimates.

9. RAPM captures comprehensive impact but cannot explain why a player is valuable or how they would perform in different contexts.

10. State-of-the-art metrics combine RAPM with box scores and tracking data for optimal estimates.

The methods introduced in this chapter form the foundation for modern player evaluation in the NBA. While specific implementations vary, the core concepts of regression-based isolation and regularization for stability remain central to every serious impact metric.

References

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3. Sill, J. (2010). Improved NBA Adjusted +/- Using Regularization and Out-of-Sample Testing. MIT Sloan Sports Analytics Conference.

4. Engelmann, J. (2017). Regularized Adjusted Plus-Minus. In Basketball Analytics (ed. K. Pape), Chapter 12. CRC Press.

5. Jacobs, D. & Silver, N. (2019). Introducing RAPTOR, Our New Metric For The Modern NBA. FiveThirtyEight.

6. Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning, 2nd ed. Springer.

7. Murphy, K. P. (2012). Machine Learning: A Probabilistic Perspective. MIT Press.

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