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Miha Kralj
86fe32a682
Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com> Co-authored-by: aider (openrouter/anthropic/claude-sonnet-4) <aider@aider.chat> Co-authored-by: Warp <agent@warp.dev>
115 lines
4.0 KiB
Markdown
115 lines
4.0 KiB
Markdown
# R²: Coefficient of Determination
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> "R² tells you how much of the variance in actual values is explained by your predictions. It's the statistician's favorite metric for good reason."
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The Coefficient of Determination (R²) measures the proportion of variance in the actual values that is predictable from the predicted values. R² ranges from negative infinity to 1, where 1 indicates perfect predictions.
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## Architecture & Physics
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R² is computed as 1 minus the ratio of residual sum of squares (RSS) to total sum of squares (TSS). This is mathematically equivalent to R² = 1 - RSE, making R² the complement of Relative Squared Error.
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### Interpretation Guide
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| R² Value | Interpretation |
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| :------- | :------------- |
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| **R² = 1** | Perfect predictions (all variance explained) |
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| **R² > 0.9** | Excellent model |
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| **R² > 0.7** | Good model |
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| **R² > 0.5** | Moderate model |
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| **R² = 0** | Model is no better than predicting the mean |
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| **R² < 0** | Model is worse than predicting the mean |
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## Mathematical Foundation
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### 1. Residual Sum of Squares (RSS)
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$$\text{RSS} = \sum_{t=1}^{n} (y_t - \hat{y}_t)^2$$
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### 2. Total Sum of Squares (TSS)
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$$\text{TSS} = \sum_{t=1}^{n} (y_t - \bar{y})^2$$
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where $\bar{y}$ is the rolling mean of actual values.
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### 3. Coefficient of Determination
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$$R^2 = 1 - \frac{\text{RSS}}{\text{TSS}} = 1 - \frac{\sum_{t=1}^{n} (y_t - \hat{y}_t)^2}{\sum_{t=1}^{n} (y_t - \bar{y})^2}$$
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### 4. Relationship to RSE
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$$R^2 = 1 - \text{RSE}$$
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## Performance Profile
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| Metric | Score | Notes |
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| :----- | :---- | :---- |
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| **Throughput** | ~40 ns/bar | Three running sums maintained |
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| **Allocations** | 0 | Zero-allocation implementation |
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| **Complexity** | O(1) | Constant time per update |
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| **Accuracy** | 10/10 | Standard statistical measure |
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| **Timeliness** | 7/10 | Rolling window introduces lag |
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| **Sensitivity** | 8/10 | Sensitive to outliers (squared errors) |
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## Common Pitfalls
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### Flat Series Problem
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When all actual values in the window are identical, TSS becomes zero (all values equal the mean). The implementation returns 0.0 in this case, indicating no variance to explain.
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### Negative R² Values
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R² can be negative when predictions are worse than simply predicting the mean. This indicates a fundamentally flawed model that should not be used.
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### R² ≠ Correlation Squared (in general)
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While R² equals the square of Pearson correlation for simple linear regression, this relationship does not hold for general predictions. R² can be negative; correlation squared cannot.
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### High R² Doesn't Mean Good Predictions
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R² measures relative fit, not absolute accuracy. A model with R² = 0.99 could still have large absolute errors if the data has high variance.
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## Usage
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```csharp
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// Create R² calculator with period 14
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var rsquared = new Rsquared(14);
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// Stream values
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var result = rsquared.Update(actual, predicted);
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Console.WriteLine($"R²: {result.Value:F4}");
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// R² > 0 = better than mean, R² = 1 = perfect
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// Batch calculation
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var r2Series = Rsquared.Calculate(actualSeries, predictedSeries, 14);
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// Zero-allocation span version
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Rsquared.Batch(actualSpan, predictedSpan, outputSpan, 14);
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```
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## R² Quick Reference
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| R² Value | Quality | Description |
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| :------- | :------ | :---------- |
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| 1.00 | Perfect | Model explains all variance |
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| 0.95 | Excellent | Model explains 95% of variance |
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| 0.80 | Good | Model explains 80% of variance |
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| 0.50 | Moderate | Model explains 50% of variance |
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| 0.00 | Poor | Model is no better than mean |
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| -0.50 | Useless | Model is worse than mean |
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## Comparison with RSE
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| Property | R² | RSE |
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| :------- | :- | :-- |
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| **Range** | (-∞, 1] | [0, +∞) |
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| **Perfect score** | 1 | 0 |
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| **Mean predictor** | 0 | 1 |
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| **Interpretation** | Variance explained | Error ratio |
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| **Relationship** | R² = 1 - RSE | RSE = 1 - R² |
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## When to Use R²
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* **Use R²** when you want an intuitive measure of model quality (0-1 scale for good models)
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* **Use RSE** when you want to compare error magnitudes directly
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* **Use both** to get complementary perspectives on model performance
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