QuanTAlib/lib/errors/mase/Mase.md

MASE: Mean Absolute Scaled Error

"A good forecast is one that's better than guessing. MASE tells you exactly how much better."

Mean Absolute Scaled Error (MASE) normalizes forecast errors by the average error of a naive "random walk" forecast (using the previous value as the prediction). This makes MASE scale-independent and interpretable across different time series.

Architecture & Physics

MASE computes a ratio: the mean absolute error of your predictions divided by the mean absolute error of a naive forecast. The naive forecast simply predicts that tomorrow's value equals today's value.

Interpretation Guide

MASE Value	Interpretation
MASE < 1	Forecast is better than naive (good)
MASE = 1	Forecast equals naive performance
MASE > 1	Forecast is worse than naive (bad)
MASE = 0	Perfect forecast

The naive baseline captures the inherent "forecastability" of the series. A highly volatile series has a larger naive error, making a given absolute error less significant.

Mathematical Foundation

1. Absolute Error

e_t = |y_t - \hat{y}_t|

2. Naive Forecast Scale

\text{Scale} = \frac{1}{n-1} \sum_{i=2}^{n} |y_i - y_{i-1}|

The scale represents the average absolute change from one period to the next.

3. Mean Absolute Scaled Error

\text{MASE} = \frac{\frac{1}{n} \sum_{t=1}^{n} |y_t - \hat{y}_t|}{\frac{1}{n-1} \sum_{i=2}^{n} |y_i - y_{i-1}|}

Or more simply:

\text{MASE} = \frac{\text{MAE}}{\text{Scale}}

Performance Profile

Metric	Score	Notes
Throughput	~35 ns/bar	Dual running sums for error and scale
Allocations	0	Zero-allocation implementation
Complexity	O(1)	Constant time per update
Accuracy	9/10	Handles edge cases well
Timeliness	7/10	Rolling window introduces lag
Robustness	10/10	Works with zero/negative values

Common Pitfalls

Flat Series Problem

When the actual series is constant (no change between values), the scale becomes zero. The implementation handles this by returning the raw MAE when scale is near zero.

Initial Warmup

The scale calculation requires at least two values (to compute differences). During warmup, MASE defaults to MAE / 1.0.

Different from Other Scaled Metrics

Unlike MAPE which scales by actual values, MASE scales by the difficulty of the forecasting problem itself.

Usage

// Create MASE calculator with period 14
var mase = new Mase(14);

// Stream values
var result = mase.Update(actual, predicted);
Console.WriteLine($"MASE: {result.Value:F4}");
// MASE < 1 = better than naive, MASE > 1 = worse than naive

// Batch calculation
var maseSeries = Mase.Calculate(actualSeries, predictedSeries, 14);

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

Comparison with Other Error Metrics

Metric	Scale-Independent	Handles Zero	Symmetric	Interpretable
MASE	✅	✅	✅	✅ (vs naive)
MAPE	✅	❌	❌	✅ (% error)
SMAPE	✅	⚠️	✅	⚠️ (bounded %)
MAE	❌	✅	✅	❌ (raw units)
RMSE	❌	✅	✅	❌ (raw units)

MASE is particularly valuable when:

• Comparing forecasts across different series
• Evaluating against a natural baseline (naive forecast)
• Working with data that includes zeros
• Needing symmetric treatment of over/under predictions