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Miha Kralj
141 lines
5.5 KiB
Markdown
141 lines
5.5 KiB
Markdown
# RSE: Relative Squared Error
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> *The squared error version of RAE. RSE and R² are two sides of the same coin: R² = 1 - RSE.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Error Metric |
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| **Inputs** | Actual vs Predicted (dual input) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (Rse) |
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| **Output range** | $\geq 0$ |
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| **Warmup** | `period` bars |
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| **PineScript** | [rse.pine](rse.pine) |
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- Relative Squared Error (RSE) measures the total squared error of predictions relative to the total squared error of a simple baseline predictor tha...
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- **Similar:** [RMSE](../rmse/Rmse.md), [Rsquared](../rsquared/Rsquared.md) | **Trading note:** Relative Squared Error; normalized by variance of actuals. >1 = worse than mean prediction.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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Relative Squared Error (RSE) measures the total squared error of predictions relative to the total squared error of a simple baseline predictor that always predicts the mean. RSE is directly related to the coefficient of determination (R²).
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## Architecture & Physics
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RSE computes a ratio of summed squared errors. The numerator is the residual sum of squares (RSS). The denominator is the total sum of squares (TSS). The relationship R² = 1 - RSE provides a direct conversion between the two metrics.
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### Interpretation Guide
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| RSE Value | R² Value | Interpretation |
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| :-------- | :------- | :------------- |
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| **RSE = 0** | **R² = 1** | Perfect predictions |
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| **RSE < 1** | **R² > 0** | Better than mean predictor |
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| **RSE = 1** | **R² = 0** | Same as mean predictor |
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| **RSE > 1** | **R² < 0** | Worse than mean predictor |
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Squared errors penalize large errors more heavily than small ones, making RSE more sensitive to outliers than RAE.
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## Mathematical Foundation
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### 1. Squared Error (RSS)
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$$e_t^2 = (y_t - \hat{y}_t)^2$$
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### 2. Squared Baseline Error (TSS)
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$$b_t^2 = (y_t - \bar{y})^2$$
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where $\bar{y}$ is the rolling mean of actual values.
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### 3. Relative Squared Error
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$$\text{RSE} = \frac{\sum_{t=1}^{n} (y_t - \hat{y}_t)^2}{\sum_{t=1}^{n} (y_t - \bar{y})^2} = \frac{\text{RSS}}{\text{TSS}}$$
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### 4. Relationship to R²
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$$R^2 = 1 - \text{RSE}$$
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## Performance Profile
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### Operation Count (Streaming Mode)
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O(1) per bar. Single-pass scalar transformation of (actual, forecast) pair; no lookback window required.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Error computation (subtract, abs/square/log) | 1-3 | ~3-8 cy | ~5-15 cy |
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| Running accumulator update (EMA or sum) | 1 | ~4 cy | ~4 cy |
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| **Total** | **2-4** | — | **~9-19 cycles** |
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Streaming update requires only the current actual/forecast pair and running state. ~10-15 cycles/bar typical.
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### Batch Mode (SIMD Analysis)
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| Operation | Vectorizable? | Notes |
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| :--- | :---: | :--- |
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| Element-wise error computation | Yes | Independent per bar; fully vectorizable with `Vector<double>` |
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| Reduction (sum/mean) | Yes | Parallel reduction; AVX2 gives 4x speedup |
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| Log/exp components | Partial | Transcendental ops; polynomial approx for SIMD |
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Batch SIMD: 4x-8x speedup for large windows. ~3-5 cy/bar amortized in vectorized batch mode.
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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** | 9/10 | 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 1.0 in this case.
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### Outlier Sensitivity
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Because errors are squared, a single large error can dominate the RSE calculation. For outlier-robust alternatives, consider RAE (which uses absolute errors).
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### Negative R² is Possible
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When RSE > 1, the implied R² is negative. This indicates predictions are worse than simply predicting the mean: a sign of a fundamentally flawed model.
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## Usage
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```csharp
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// Create RSE calculator with period 14
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var rse = new Rse(14);
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// Stream values
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var result = rse.Update(actual, predicted);
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Console.WriteLine($"RSE: {result.Value:F4}");
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Console.WriteLine($"Implied R²: {1 - result.Value:F4}");
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// RSE < 1 = better than mean, R² > 0
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// Batch calculation
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var rseSeries = Rse.Calculate(actualSeries, predictedSeries, 14);
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// Zero-allocation span version
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Rse.Batch(actualSpan, predictedSpan, outputSpan, 14);
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```
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## RSE vs R² Quick Reference
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| Scenario | RSE | R² | Quality |
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| :------- | :-- | :- | :------ |
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| Perfect model | 0.00 | 1.00 | Excellent |
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| Very good model | 0.05 | 0.95 | Very good |
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| Good model | 0.20 | 0.80 | Good |
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| Moderate model | 0.50 | 0.50 | Moderate |
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| Poor model (= mean) | 1.00 | 0.00 | Poor |
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| Useless model | 2.00 | -1.00 | Useless |
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## Comparison with RAE
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| Property | RSE | RAE |
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| :------- | :-- | :-- |
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| **Error type** | Squared (L2) | Absolute (L1) |
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| **Outlier sensitivity** | High | Low |
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| **Related to** | R² | — |
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| **Baseline** | Mean predictor | Mean predictor |
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| **Interpretation** | 1 - R² | Better/worse than mean | |