QuanTAlib/lib/errors/rse/Rse.md

RSE: Relative Squared Error

"The squared error version of RAE. RSE and R² are two sides of the same coin: R² = 1 - RSE."

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).

Architecture & Physics

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.

Interpretation Guide

RSE Value	R² Value	Interpretation
RSE = 0	R² = 1	Perfect predictions
RSE < 1	R² > 0	Better than mean predictor
RSE = 1	R² = 0	Same as mean predictor
RSE > 1	R² < 0	Worse than mean predictor

Squared errors penalize large errors more heavily than small ones, making RSE more sensitive to outliers than RAE.

Mathematical Foundation

1. Squared Error (RSS)

e_t^2 = (y_t - \hat{y}_t)^2

2. Squared Baseline Error (TSS)

b_t^2 = (y_t - \bar{y})^2

where \bar{y} is the rolling mean of actual values.

3. Relative Squared Error

\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}}

4. Relationship to R²

R^2 = 1 - \text{RSE}

Performance Profile

Metric	Score	Notes
Throughput	~40 ns/bar	Three running sums maintained
Allocations	0	Zero-allocation implementation
Complexity	O(1)	Constant time per update
Accuracy	9/10	Standard statistical measure
Timeliness	7/10	Rolling window introduces lag
Sensitivity	8/10	Sensitive to outliers (squared errors)

Common Pitfalls

Flat Series Problem

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.

Outlier Sensitivity

Because errors are squared, a single large error can dominate the RSE calculation. For outlier-robust alternatives, consider RAE (which uses absolute errors).

Negative R² is Possible

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.

Usage

// Create RSE calculator with period 14
var rse = new Rse(14);

// Stream values
var result = rse.Update(actual, predicted);
Console.WriteLine($"RSE: {result.Value:F4}");
Console.WriteLine($"Implied R²: {1 - result.Value:F4}");
// RSE < 1 = better than mean, R² > 0

// Batch calculation
var rseSeries = Rse.Calculate(actualSeries, predictedSeries, 14);

// Zero-allocation span version
Rse.Batch(actualSpan, predictedSpan, outputSpan, 14);

RSE vs R² Quick Reference

Scenario	RSE	R²	Quality
Perfect model	0.00	1.00	Excellent
Very good model	0.05	0.95	Very good
Good model	0.20	0.80	Good
Moderate model	0.50	0.50	Moderate
Poor model (= mean)	1.00	0.00	Poor
Useless model	2.00	-1.00	Useless

Comparison with RAE

Property	RSE	RAE
Error type	Squared (L2)	Absolute (L1)
Outlier sensitivity	High	Low
Related to	R²	—
Baseline	Mean predictor	Mean predictor
Interpretation	1 - R²	Better/worse than mean