Evaluating Football (Soccer) Players

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# Penalized Regression in Action: Evaluating Football (Soccer) Players using Event Stream Data

.large[Robert Bajons | Joint work with Kurt Hornik | 5 Oct. 2023]

Vienna University of Economics and Business 
Austrian Statistic Days 2023

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# Motivation

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# Evaluating Players in Football

- Classical approach: Use count based statistics (goals, assists, pass completion, key passes).

**How can evaluation be done more sophisticated?**

- Advances in collecting data lead to more data driven approaches:

- **Bottom-Up rating systems:** assign values to each action performed `$\Rightarrow$` aggregate values for each player over course of a relevant period (i.e. a match or a season).
 
 - Examples: expected Goal (xG), expected Assists (xA), Valuing Players by Estimating Probabilities (VAEP).

--
 - **Top-Down rating systems:** break down the whole team performance `$\Rightarrow$` distribute credit onto players involved.
 - Examples: Plus Minus (PM) ratings.

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# Data and Preprocessing

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# Event Stream Data

**Raw Data:** Information on every action happening during a match (pass, shots, fouls, etc.). Use data from 2017/18 season of Spanish La Liga.

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# Data Preprocessing

- Build a data set of **possession sequences**:

- Sequence of consecutive actions ended by either opponent team gaining possession or stopped by the referee.

- Extract important information from the data:

- **On-ball information:** Which player was directly involved in the possession (binary).
 
 - **Off-ball information:** Which players was on the field (attacking team = 1, defensive team = -1, not on field = 0).
 
--

- Model matrix ( `$N = N_{inv}+N_{of}$` columns, `$N_p$` rows): 
`$$X_{mod} = (X_{N_{inv}},X_{N_{of}})$$`

--
  
- Response Variable:
  - Possession ends with goal or not (binary).

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# Methodology
  
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# Player Evaluation Problem

- Derive a rating from measuring a players contribution to (successful) possession.

- .my-style[First problem:] Two types of contribution: direct (on-ball), indirect (off-ball). 
  
--
  
- Naive approach: fit (generalized) linear model on possession data and interpret model coefficients `$\beta$`.

- .my-style[Second problem:] High dimensionality, imbalanced data setup `$\Rightarrow$` Multicollinearity, separable data, overfitting.

- **Solution:** Add regularization term to estimation of coefficients (and cross validation).

- **Question:** Which type of regularization is appropriate two solve both of the above problems?

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# Ridge Regression

- The ridge estimator:

`$$\hat{\beta}_{ridge} = \underset{\beta}{\operatorname{argmin}} \text{ } \ell(Y;X,\beta) + \lambda ||\beta||^2_2,$$`
  `$\ell(Y;X,\beta)$` (negative) log-likelihood function for binomial logistic regression model.

- **Implications:** 
  - Multicollinearty and overfitting may be avoided/reduced.
  - Players effects are shrunken towards 0. 
  - Strong (resp weak) players obtain less shrinkage.
  - Players of similar strength are shrunken to a similar value. 
  - Each player receives a strength estimate (distinction from lasso penalty).
  - Distinction between direct (on-ball) and indirect (off-ball) effects possible.
  
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# Other penalization structures

**Can we do "even better" than ridge regression? What about different penalization structures?**

- Lasso penalty? .my-style[Disadvantage:] A lot of players set exactly to zero (little interpretability, bad for rating of players).

What we desire:

- Rating for each player (or at least a group of the players).

- Differentiation between shrinkage on direct and indirect effects.

- Grouping of players into prespecified groups `$\Rightarrow$` **Group lasso, exclusive lasso**

- Grouping of players into similar groups `$\Rightarrow$` **Generalized lasso**

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# Group Lasso

- The group lasso estimator:

`$$\hat{\beta}_{gl} = \underset{\beta}{\operatorname{argmin}} \text{ } \ell(Y;X,\beta)+\lambda \sum_{l=1}^m \sqrt{p_l}\left\|\beta^{(l)}\right\|_2.$$`
 `$m$` number of groups, `$p_l$` size of group `$l$` (see e.g. <a name=cite-SGL13></a>[Simon, Friedman, Hastie, and Tibshirani (2013)](#bib-SGL13)).

- **Implications:**

- On group level: similar to lasso `$\Rightarrow$` sets whole groups to 0.
  - Within group level: ridge-like penalty `$\Rightarrow$` similar implications as before but within group.

- **Application**

- Natural grouping of players into (at least) 4 position groups: Forwards, midfielders, defenders and goalkeepers.
  - Grouping of direct and indirect effects possible (i.e. in total 8 groups).
  - Distinction between groups highly desired: tasks on field varies for each position group.
  
  
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# Exclusive Lasso

- The group lasso estimator:

`$$\hat{\beta}_{excl} = \underset{\beta}{\operatorname{argmin}} \text{ } \ell(Y;X,\beta)+\frac{\lambda}{2} \sum_{l=1}^m \left\|\beta^{(l)}\right\|_1^2.$$`
 `$m$` number of groups (see e.g. <a name=cite-ExL17></a>[Campbell and Allen (2017)](#bib-ExL17)).

- **Implications:**

- On group level: similar to ridge `$\Rightarrow$` in each group at least one non-zero variable.
  - Within group level: lasso-like penalty `$\Rightarrow$` only strong (resp weak) players obtain coefficient different from 0.

- **Application**

- Especially interesting for scouting: penalty allows for selecting strongest player of each group.
  - Consider e.g.: Team is interested in player on particular position, not exceeding a certain age level, affordable (market value below certain threshold).
  - Comparison to other groups still relevant.

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# Generalized Lasso

- The group lasso estimator:

`$$\hat{\beta}_{genl} = \underset{\beta}{\operatorname{argmin}} \text{ } \ell(Y;X,\beta)+\lambda ||D\beta||_1,$$`
 `$D$` matrix that allows for individual specification of lasso type penalties (see e.g. <a name=cite-GenLasso11></a>[Tibshirani and Taylor (2011)](#bib-GenLasso11)). 
 
- Special case: Ranking lasso (<a name=cite-RL12></a>[Masarotto and Varin (2012)](#bib-RL12))

`$$\hat{\beta}_{rl} = \underset{\beta}{\operatorname{argmin}} \text{ } \ell(Y;X,\beta)+\lambda \sum_{i<j}^N w_{ij}|\beta_i-\beta_j|.$$`
 
- **Application:**

- Apply lasso penalty on the differences of player strengths (ranking lasso).
  - Set up matrix `$D$`, such that is divided into part related direct and indirect estimates. 
  - Goal: Obtain strength groups of players, without employing a prespecified grouping.

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# Results
  
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# Ridge Ratings

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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ad925864{width:1.5in;background-color:rgba(52, 59, 232, 1.00);vertical-align: middle;border-bottom: 1pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ad925865{width:1.5in;background-color:rgba(255, 193, 37, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 1pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ad92586c{width:1.5in;background-color:rgba(255, 193, 37, 1.00);vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ad92586d{width:1.5in;background-color:rgba(255, 193, 37, 1.00);vertical-align: middle;border-bottom: 1pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-ad977a68'><thead><tr style="overflow-wrap:break-word;"><th class="cl-ad925830">Position</th><th class="cl-ad925830">Player</th><th class="cl-ad92583a">Score Inv</th><th class="cl-ad92583a">Score Of</th><th class="cl-ad92583a">Score Sum</th><th class="cl-ad92583a">Rank Inv</th><th class="cl-ad92583a">Rank Of</th><th class="cl-ad92583a">Rank Sum</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-ad925844">Forward</td><td class="cl-ad925845">L. Messi</td><td class="cl-ad925846">0.04088</td><td class="cl-ad925846">0.03344</td><td class="cl-ad925846">0.07432</td><td class="cl-ad925846">1</td><td class="cl-ad925846">1</td><td class="cl-ad925846">1</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad925844">Forward</td><td class="cl-ad925845">L. Suárez</td><td class="cl-ad925846">0.03545</td><td class="cl-ad925846">0.02547</td><td class="cl-ad925846">0.06092</td><td class="cl-ad925846">2</td><td class="cl-ad925846">3</td><td class="cl-ad925846">2</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad92584e">Forward</td><td class="cl-ad92584f">Cristiano Ronaldo</td><td class="cl-ad925850">0.03228</td><td class="cl-ad925850">0.01284</td><td class="cl-ad925850">0.04511</td><td class="cl-ad925850">3</td><td class="cl-ad925850">44</td><td class="cl-ad925850">3</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad925851">Midfielder</td><td class="cl-ad925858">Paulinho</td><td class="cl-ad925859">0.00942</td><td class="cl-ad925859">0.02667</td><td class="cl-ad925859">0.03609</td><td class="cl-ad925859">40</td><td class="cl-ad925859">2</td><td class="cl-ad925859">5</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad92585a">Midfielder</td><td class="cl-ad925845">I. Rakitić</td><td class="cl-ad925846">0.01143</td><td class="cl-ad925846">0.02391</td><td class="cl-ad925846">0.03534</td><td class="cl-ad925846">32</td><td class="cl-ad925846">8</td><td class="cl-ad925846">6</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad92585b">Midfielder</td><td class="cl-ad92584f">Pablo Fornals</td><td class="cl-ad925850">0.01553</td><td class="cl-ad925850">0.01568</td><td class="cl-ad925850">0.03121</td><td class="cl-ad925850">17</td><td class="cl-ad925850">31</td><td class="cl-ad925850">10</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad925862">Defender</td><td class="cl-ad925858">Nacho</td><td class="cl-ad925859">0.00822</td><td class="cl-ad925859">0.02456</td><td class="cl-ad925859">0.03278</td><td class="cl-ad925859">56</td><td class="cl-ad925859">5</td><td class="cl-ad925859">8</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad925863">Defender</td><td class="cl-ad925845">Piqué</td><td class="cl-ad925846">0.00758</td><td class="cl-ad925846">0.02445</td><td class="cl-ad925846">0.03204</td><td class="cl-ad925846">62</td><td class="cl-ad925846">6</td><td class="cl-ad925846">9</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad925864">Defender</td><td class="cl-ad92584f">Marcelo</td><td class="cl-ad925850">0.00707</td><td class="cl-ad925850">0.02192</td><td class="cl-ad925850">0.02899</td><td class="cl-ad925850">66</td><td class="cl-ad925850">11</td><td class="cl-ad925850">13</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad925865">Goalkeeper</td><td class="cl-ad925858">M. ter Stegen</td><td class="cl-ad925859">0.00365</td><td class="cl-ad925859">0.02344</td><td class="cl-ad925859">0.02710</td><td class="cl-ad925859">105</td><td class="cl-ad925859">9</td><td class="cl-ad925859">15</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad92586c">Goalkeeper</td><td class="cl-ad925845">K. Navas</td><td class="cl-ad925846">0.00607</td><td class="cl-ad925846">0.01682</td><td class="cl-ad925846">0.02289</td><td class="cl-ad925846">73</td><td class="cl-ad925846">25</td><td class="cl-ad925846">24</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ad92586d">Goalkeeper</td><td class="cl-ad92584f">J. Oblak</td><td class="cl-ad925850">-0.00055</td><td class="cl-ad925850">0.02188</td><td class="cl-ad925850">0.02133</td><td class="cl-ad925850">305</td><td class="cl-ad925850">12</td><td class="cl-ad925850">32</td></tr></tbody></table></div>

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# Ridge vs Group Lasso

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# Generalized Lasso

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# Evaluating Player Ratings
  
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# Testing the Validity of a Rating

- **<a name=cite-HvattumGelade21></a>[Hvattum and Gelade (2021)](#bib-HvattumGelade21):**

*If ratings are accurate in assessing the capabilities of players, one should be able to accurately predict the outcome of a match based only on the ratings of the players involved.*
  
--

- Use match outcome data of the 2017/18 season.

- Predict match results via two state of the art models:

- Bivariate Poisson model (<a name=cite-KN03></a>[Karlis and Ntzoufras (2003)](#bib-KN03)).
 - Ordered logistic regression (<a name=cite-ArntzenHvattum21></a>[Arntzen and Hvattum (2021)](#bib-ArntzenHvattum21)).
 
- Use average player rating differences between teams as covariates.

- Evaluate predictive performances of the models with covariates using proper scoring rules (Brier score (BS), informational loss (IL)).

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# Validity Results

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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ae64b49e{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 1pt solid rgba(102, 102, 102, 1.00);border-left: 1pt solid rgba(102, 102, 102, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ae64b4a6{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 1pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 1pt solid rgba(102, 102, 102, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ae64b4a7{width:1.5in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 1pt solid rgba(102, 102, 102, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ae64b4a8{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 1pt solid rgba(102, 102, 102, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ae64b4b0{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 1pt solid rgba(102, 102, 102, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-ae64b4b1{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-ae692dec'><thead><tr style="overflow-wrap:break-word;"><th class="cl-ae64b41a"></th><th colspan="2"class="cl-ae64b424">Ordinal Logistic</th><th colspan="2"class="cl-ae64b424">Bivariate Poisson</th></tr><tr style="overflow-wrap:break-word;"><th class="cl-ae64b41a">Strength Model</th><th class="cl-ae64b424">BS</th><th class="cl-ae64b42e">IL</th><th class="cl-ae64b424">BS</th><th class="cl-ae64b42f">IL</th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-ae64b430">Ridge Sum</td><td class="cl-ae64b438">0.576</td><td class="cl-ae64b439">1.398</td><td class="cl-ae64b438">0.581</td><td class="cl-ae64b43a">1.406</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b442">Ridge Avg</td><td class="cl-ae64b47e">0.574</td><td class="cl-ae64b488">1.392</td><td class="cl-ae64b47e">0.579</td><td class="cl-ae64b489">1.403</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b48a">Group Lasso Sum</td><td class="cl-ae64b492">0.566</td><td class="cl-ae64b493">1.381</td><td class="cl-ae64b492">0.578</td><td class="cl-ae64b49c">1.404</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b442">Group Lasso Avg</td><td class="cl-ae64b47e">0.569</td><td class="cl-ae64b488">1.387</td><td class="cl-ae64b47e">0.580</td><td class="cl-ae64b49d">1.410</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b48a">Excl Lasso Sum</td><td class="cl-ae64b49e">0.577</td><td class="cl-ae64b4a6">1.401</td><td class="cl-ae64b49e">0.589</td><td class="cl-ae64b49c">1.427</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b442">Excl Lasso Avg</td><td class="cl-ae64b47e">0.577</td><td class="cl-ae64b488">1.403</td><td class="cl-ae64b47e">0.589</td><td class="cl-ae64b49d">1.427</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b48a">Gen Lasso Sum</td><td class="cl-ae64b49e">0.611</td><td class="cl-ae64b4a6">1.470</td><td class="cl-ae64b49e">0.618</td><td class="cl-ae64b49c">1.482</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b442">Gen Lasso Avg</td><td class="cl-ae64b47e">0.596</td><td class="cl-ae64b488">1.443</td><td class="cl-ae64b47e">0.608</td><td class="cl-ae64b49d">1.465</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b48a">PCV</td><td class="cl-ae64b49e">0.631</td><td class="cl-ae64b4a6">1.514</td><td class="cl-ae64b49e">0.632</td><td class="cl-ae64b49c">1.514</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b430">ELO</td><td class="cl-ae64b438">0.638</td><td class="cl-ae64b439">1.525</td><td class="cl-ae64b438">0.614</td><td class="cl-ae64b43a">1.466</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-ae64b4a7">Baseline</td><td class="cl-ae64b4a8">0.636</td><td class="cl-ae64b4b0">1.521</td><td class="cl-ae64b4a8">0.637</td><td class="cl-ae64b4b1">1.523</td></tr></tbody></table></div>

---
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# Thank You!

Contact: [robert.bajons@wu.ac.at](mailto:robert.bajons@wu.ac.at)

(Don't forget to scan the QR-code <svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:darkgreen;overflow:visible;position:relative;"><path d="M435.4 361.3l-89.7-6c-5.2-.3-10.3 1.1-14.5 4.2s-7.2 7.4-8.4 12.5l-22 87.2c-14.4 3.2-29.4 4.8-44.8 4.8s-30.3-1.7-44.8-4.8l-22-87.2c-1.3-5-4.3-9.4-8.4-12.5s-9.3-4.5-14.5-4.2l-89.7 6C61.7 335.9 51.9 307 49 276.2L125 228.3c4.4-2.8 7.6-7 9.2-11.9s1.4-10.2-.5-15L100.4 118c19.9-22.4 44.6-40.5 72.4-52.7l69.1 57.6c4 3.3 9 5.1 14.1 5.1s10.2-1.8 14.1-5.1l69.1-57.6c27.8 12.2 52.5 30.3 72.4 52.7l-33.4 83.4c-1.9 4.8-2.1 10.1-.5 15s4.9 9.1 9.2 11.9L463 276.2c-3 30.8-12.7 59.7-27.6 85.1zM256 48l.9 0h-1.8l.9 0zM56.7 196.2c.9-3 1.9-6.1 2.9-9.1l-2.9 9.1zM132 423l3.8 2.7c-1.3-.9-2.5-1.8-3.8-2.7zm248.1-.1c-1.3 1-2.7 2-4 2.9l4-2.9zm75.2-226.6l-3-9.2c1.1 3 2.1 6.1 3 9.2zM256 512A256 256 0 1 0 256 0a256 256 0 1 0 0 512zm14.1-325.7c-8.4-6.1-19.8-6.1-28.2 0L194 221c-8.4 6.1-11.9 16.9-8.7 26.8l18.3 56.3c3.2 9.9 12.4 16.6 22.8 16.6h59.2c10.4 0 19.6-6.7 22.8-16.6l18.3-56.3c3.2-9.9-.3-20.7-8.7-26.8l-47.9-34.8z"/></svg>)

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