QuanTAlib/lib/errors/rmsle/Rmsle.md

RMSLE: Root Mean Squared Logarithmic Error

"RMSLE: because sometimes your errors need to be measured in decades, not dollars."

Root Mean Squared Logarithmic Error is the square root of MSLE, providing an error metric in log-scale units. This makes RMSLE more interpretable than MSLE while retaining all its benefits for data spanning multiple orders of magnitude.

Architecture & Physics

RMSLE computes the root mean of squared log differences:

\text{RMSLE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} \left(\log(1 + \text{actual}_i) - \log(1 + \text{predicted}_i)\right)^2}

The relationship to MSLE is straightforward:

\text{RMSLE} = \sqrt{\text{MSLE}}

Interpretability

RMSLE values correspond directly to log-scale error:

• RMSLE = 0.1 → approximately 10% ratio error
• RMSLE = 0.69 → approximately 100% ratio error (2:1 or 1:2 ratio)
• RMSLE = 1.0 → approximately 170% ratio error (~2.7:1 ratio)

Mathematical Foundation

1. Log Transform

\tilde{x} = \log(1 + x)

2. Root Mean Square in Log Space

\text{RMSLE} = \sqrt{\frac{1}{n} \sum_{i=t-n+1}^{t} \left(\tilde{\text{actual}}_i - \tilde{\text{predicted}}_i\right)^2}

3. Approximation for Small Errors

For small relative errors (\epsilon):

\text{RMSLE} \approx |\log(1 + \epsilon)| \approx |\epsilon|

Performance Profile

Metric	Score	Notes
Throughput	28 ns/bar	O(1) with sqrt overhead
Allocations	0	Zero-allocation hot path
Complexity	O(1)	Constant per update
Outlier Robustness	9/10	Log compression
Interpretability	7/10	Better than MSLE
Scale Independence	10/10	Ratio-based
Zero Handling	10/10	Uses 1+x transform

Usage

// Streaming mode - track prediction quality
var rmsle = new Rmsle(20);

// Revenue predictions across different scales
rmsle.Update(actual: 1000.0, predicted: 950.0);     // Small business
rmsle.Update(actual: 1000000.0, predicted: 950000.0); // Enterprise

double logError = rmsle.Last.Value;
Console.WriteLine($"RMSLE: {logError:F3}"); // Consistent ~0.05 for 5% error

// Batch mode - backtest analysis
var actual = new TSeries { 100, 1000, 10000, 100000 };
var predicted = new TSeries { 95, 950, 9500, 95000 };
var results = Rmsle.Calculate(actual, predicted, period: 3);

// Span mode - zero-allocation bulk processing
Span<double> output = stackalloc double[1000];
Rmsle.Batch(actualSpan, predictedSpan, output, period: 20);

Interpretation Guide

RMSLE Value	Interpretation	Typical Application
< 0.1	Excellent	High-precision forecasting
0.1 - 0.3	Good	Business forecasting
0.3 - 0.5	Moderate	General ML models
0.5 - 1.0	Poor	Needs improvement
> 1.0	Very poor	Model redesign needed

Converting RMSLE to Ratio Error

\text{Typical Ratio} \approx e^{\text{RMSLE}}

RMSLE	Ratio Factor	Meaning
0.1	1.105	Predictions typically within ±10.5%
0.2	1.221	Predictions typically within ±22%
0.5	1.649	Predictions typically within ±65%
0.693	2.0	Predictions off by factor of 2
1.0	2.718	Predictions off by factor of e

Comparison: RMSE vs RMSLE

var rmse = new Rmse(1);
var rmsle = new Rmsle(1);

// Small scale
rmse.Update(100.0, 50.0);     // RMSE = 50
rmsle.Update(100.0, 50.0);    // RMSLE ≈ 0.69

// Large scale (same ratio)
rmse.Update(1000000.0, 500000.0);   // RMSE = 500,000
rmsle.Update(1000000.0, 500000.0);  // RMSLE ≈ 0.69

// RMSE varies wildly; RMSLE is consistent for same ratio

Use Cases

1. E-Commerce Sales Forecasting

Product sales vary from single units to thousands:

// Product A: sells 5 units, predicted 4
// Product B: sells 5000 units, predicted 4000
// Same 20% under-prediction, similar RMSLE

2. Financial Modeling

Stock prices, market caps, and volumes span many magnitudes:

// Penny stock: $0.10 → $0.12 (20% move)
// Blue chip: $100 → $120 (20% move)
// RMSLE treats these equivalently

3. Scientific Measurements

Population counts, concentrations, or any log-normal data:

// Bacteria count: 1,000 → 1,200
// Bacteria count: 1,000,000,000 → 1,200,000,000
// Same relative accuracy

Common Pitfalls

1. Non-Negative Requirement

RMSLE requires both actual and predicted values to be non-negative:

// Invalid inputs are replaced with last valid value or 0
rmsle.Update(-100.0, 50.0);  // Uses last valid actual

2. Unit Interpretation

RMSLE is in "log units," not the original units:

// RMSLE = 0.5 does NOT mean $0.50 error
// It means predictions are typically off by ~65% ratio

3. Near-Zero Sensitivity

Small absolute values near zero can produce large RMSLE:

// actual=1, predicted=10: RMSLE = |log(2) - log(11)| ≈ 1.7
// actual=1000, predicted=10000: RMSLE = |log(1001) - log(10001)| ≈ 2.3
// Not exactly proportional due to 1+x offset

Relationship to Other Metrics

Metric	Relationship
MSLE	RMSLE = √MSLE
RMSE	Different scale sensitivity
MAPE	Both percentage-like, but RMSLE handles zeros
MAE	RMSLE is log-transformed, squared, then rooted

See Also

• MSLE - Squared version without root
• RMSE - Linear-scale root mean squared error
• MAPE - Percentage error without log transform