Ensemble Rating System
====================

Overview
--------

The Ensemble rating system in Elote is a meta-rating approach that combines multiple rating systems to leverage their individual strengths while mitigating their weaknesses. By aggregating predictions from different rating algorithms, the Ensemble system can potentially provide more robust and accurate predictions than any single rating system alone.

This approach is inspired by ensemble methods in machine learning, where combining multiple models often leads to better performance than any individual model. The Ensemble competitor in Elote allows you to combine any of the implemented rating systems (Elo, Glicko, ECF, DWZ) with customizable weights.

How It Works
-----------

The Ensemble rating system works by:

1. Maintaining multiple rating systems for each competitor
2. Calculating expected outcomes from each system
3. Combining these predictions using a weighted average
4. Updating each underlying rating system after matches

The expected outcome calculation is:

.. math::

   E_{ensemble} = \sum_{i=1}^{n} w_i \times E_i

Where:
- :math:`E_{ensemble}` is the ensemble expected score
- :math:`E_i` is the expected score from rating system i
- :math:`w_i` is the weight assigned to rating system i
- :math:`n` is the number of rating systems in the ensemble

After a match, each underlying rating system is updated according to its own update rules, and the ensemble prediction is recalculated.

Advantages
---------

- **Robustness**: Less sensitive to the quirks of any single rating system
- **Accuracy**: Can achieve better predictive performance by combining complementary systems
- **Flexibility**: Can be customized with different component systems and weights
- **Adaptability**: Works well across different domains and competition structures
- **Graceful Degradation**: If one system performs poorly in a specific scenario, others can compensate

Limitations
----------

- **Complexity**: More complex to implement and understand than single rating systems
- **Computational Overhead**: Requires calculating and updating multiple rating systems
- **Parameter Tuning**: Finding optimal weights may require experimentation
- **Black Box Nature**: The combined prediction may be harder to interpret
- **Cold Start**: Requires sufficient data to properly calibrate all component systems

Implementation in Elote
----------------------

Elote provides an implementation of the Ensemble rating system through the ``EnsembleCompetitor`` class:

.. code-block:: python

    from elote import EnsembleCompetitor
    from elote import EloCompetitor, GlickoCompetitor

    # Create an ensemble with Elo and Glicko components
    player1 = EnsembleCompetitor(
        rating_systems=[
            (EloCompetitor(initial_rating=1500), 0.7),
            (GlickoCompetitor(initial_rating=1500, initial_rd=350), 0.3)
        ]
    )
    
    player2 = EnsembleCompetitor(
        rating_systems=[
            (EloCompetitor(initial_rating=1600), 0.7),
            (GlickoCompetitor(initial_rating=1600, initial_rd=350), 0.3)
        ]
    )

    # Get win probability
    win_probability = player2.expected_score(player1)
    print(f"Player 2 win probability: {win_probability:.2%}")

    # Record a match result
    player1.beat(player2)  # Player 1 won!

    # All underlying ratings are automatically updated
    print(f"Player 1 ensemble expected score vs Player 2: {player1.expected_score(player2):.2%}")

Customization
------------

The ``EnsembleCompetitor`` class allows for extensive customization:

.. code-block:: python

    from elote import EnsembleCompetitor, EloCompetitor, GlickoCompetitor, ECFCompetitor, DWZCompetitor

    # Create an ensemble with all available rating systems
    player = EnsembleCompetitor(
        rating_systems=[
            (EloCompetitor(initial_rating=1500), 0.4),
            (GlickoCompetitor(initial_rating=1500), 0.3),
            (ECFCompetitor(initial_rating=120), 0.2),
            (DWZCompetitor(initial_rating=1500), 0.1)
        ]
    )

Key considerations:
- The weights should sum to 1.0 for proper probabilistic interpretation
- Higher weights give more influence to that rating system
- You can include any combination of rating systems
- Each component can be customized with its own parameters

Choosing Weights
--------------

There are several approaches to choosing weights for your ensemble:

1. **Equal Weights**: Start with equal weights for all systems
2. **Domain Knowledge**: Assign weights based on known performance in your domain
3. **Cross-Validation**: Use historical data to find optimal weights
4. **Adaptive Weights**: Dynamically adjust weights based on each system's performance

For most applications, starting with equal weights and then adjusting based on observed performance is a practical approach.

Real-World Applications
---------------------

Ensemble rating systems are valuable in:

- **Sports Analytics**: Combining multiple models for more accurate predictions
- **Game Matchmaking**: Creating balanced matches in competitive games
- **Recommendation Systems**: Ranking items based on multiple criteria
- **Tournament Design**: Seeding players based on robust ratings
- **Decision Making**: Aggregating multiple ranking methods for group decisions

References
---------

1. Dietterich, T. G. (2000). "Ensemble Methods in Machine Learning". Multiple Classifier Systems, 1-15.
2. Seni, G., & Elder, J. F. (2010). "Ensemble Methods in Data Mining: Improving Accuracy Through Combining Predictions". Synthesis Lectures on Data Mining and Knowledge Discovery, 2(1), 1-126.
3. Graepel, T., Herbrich, R., & Gold, J. (2004). "Learning to Fight". Proceedings of the International Conference on Computer Games: Artificial Intelligence, Design and Education.